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LECTURE-NOTES 



ON THE 



THEORY OF ELECTRICAL 
MEASUREMENTS. 

PREPARED FOR THE THIRD-YEAR 

CLASSES OF THE COOPER UNION 

NIGHT-SCHOOL OF SCIENCE. 



BY 

WILLIAM A. ANTHONY, 

Professor of Physics. 



FI R S T EDITION. 
FIRST THOUSAND. 



NEW YORK : 
JOHN WILEY & SONS. 
London: CHAPMAN & HALL, Limited. 
1898. 



16739 



Copyright, 1898, 

BY 

WILLIAM A. ANTHONY. 



f" 



* 



b v 




TWO COPIES RECEIVED* 



ROBERT PKUMMOND, PRINTER, NEW YORK. 

*893„ 



PREFACE. 



It is the purp ;e of these Notes to furnish the 
student the topics treated in the lectures; to give in 
full such facts, data, and courses of reasoning as are 
not readily mastered by the student in the course of 
the lecture; in short, to aid the student in those 
matters which he is likely to find most difficult, or 
which he may fail at the moment to fully grasp. It 
is not, however, intended to relieve the student of the 
necessity of taking his own notes of the lectures as 
they are delivered, and every student is expected to 
take such notes, especially as to the experiments and 
the lessons which they teach. Unless the student 
writes short-hand, the notes he can take during a 
lecture must necessarily be brief, and he should aim 
to record suggestive phrases to be afterward more 
fully elaborated. 

The student is advised of the great importance, as 
soon as possible after the lecture, of going over the 
ground and filling in his notes, thereby clearing up 



m 



IV PREFA CE. 

subjects that may have appeared obscure, and fixing 
his knowledge of the topics treated. On no account 
should the student fail to make this review before the 
time for the next lecture. 

It is the purpose of this course of lectures to teach 
the fundamental principles of all electrical measure- 
ments, rather than to describe in detail methods em- 
ployed in particular cases. To this end the definitions 
and relations of electrical quantities have been dis- 
cussed at considerable length, and the derivation of 
the electrical units has been fully treated. It is 
believed that, with this thorough foundation in the 
theory of electrical measurements, the student will be 
better equipped for practical work with a knowledge 
of typical methods such as are treated in these lec- 
tures, than he could possibly be by the most extended 
empirical instruction in special methods without such 
foundation. 



CONTENTS. 



The C. G. S. System of Units Page i 

Arbitrary and Derived Units; Basis of C. G. S. Sys- 
tem; Fundamental Units; Characteristics of Matter; 
Motion; Units of Velocity and of Acceleration; Force; 
Work and Energy; Practical Units; Gravitation Units; 
Heat Units; Relations between the Various Units. 

The Magnetic Field Page 12 

General Character of Magnets; Lines of Magnetic 
Force; Strength of Magnetic Poles; L'nit Magnetic Pole; 
Intensity of Magnetic Fields; Graphical Representation of 
Magnetic Fields; The Earth's Magnetic Field; Measure- 
ment of Field Intensity. 

The Electric Current Page 20 

Origin and Effect of Currents; Magnetic Field due to 
Currents; The Tangent Galvanometer; Current-measur- 
ing Instruments Employing Artificial Fields; Electrical 
Dynamometers and Electric Balances. 

Potential and Electromotive Force Page 29 

Potential and Difference of Potential in General; Elec- 
trical Difference of Potential; Measurement of Difference 
of Potential; Unit Difference of Potential; Electromotive 
Force; Examples of Electromotive Force. 

Resistance and Ohm's Law Page 36 

Specific Resistance; Unit Resistance; Conductivity; 
Value of the Ohm; Power Consumed by Resistance; Di- 
vided Circuits. 

Practical Measurements of Electrical Quan- 
tities ... Page 43 

The International Units; Definitions of the Interna- 
tional Units by the International Congress. 

V 



VI CONTENTS. 

Measurements of Resistance Page 45 

Instruments; Methods of Measurement; Measurement 
of Insulation Resistance; Measurement of Very Small 
Resistances; Measurement of Resistance of Eletrolytes; 
Resistance of Batteries. 

Measurement of Current Page 53 

Measurement by the Tangent Galvanometer; By Di- 
rect reading Instruments; By Fall of Potential. 

Measurement of Potential Page 55 

By Electrostatic Forces; By Comparison with a Known 
Potential Difference; By the Current Produced in a Cir- 
cuit of Known Resistance; By the Ballistic Galvanometer. 

Testing and Calibrating Instruments Page 58 

Tests of Resistance Sets; The Divided-meter Bridge; 
Calibration of the Bridge Wire; Tests of Current Instru- 
ments; Tests of Instruments for Measuring Potential. 

Heating Effects of the Current , Page 63 

Temperature of a Conductor Carrying Current. 

Incandescent Lighting Page 65 

The Incandescent Lamp; High Temperature Required 
for Economy; Incandescent Lamps of Different Candle- 
power. 

Arc Lighting Page 67 

Source of the Light; Power consumed. 

Chemical Effects of the Current Page 70 

Decomposition of Salts; Definitions of Terms; Faraday's 
Laws; Applications of Electrolysis; Energy Required. 

Electromagnetic Induction Page 76 

Direction of Current ; Law of Lenz; Electromotive 
Force Developed. 

Electromagnetism Page 79 

The Magnetic Circuit; Magnetic Reluctance and Per- 
meability; The Electromagnet; Measurements of Perme- 
ability. 



NOTES 



UPON 



ELECTRICAL MEASUREMENTS. 



THE C.G.S. SYSTEM OF UNITS. 

To measure any quantity is to compare it with 
another quantity of the same kind that has been 
chosen as a unit. 

It was the practice originally to choose units arbi- 
trarily, without regard to the relation they might 
bear to other units already chosen. Thus the gallon, 
a unit of capacity, is 231 cubic inches. It would 
have been far more convenient if it had been 100 or 
IOOO cubic inches. 

When electrical science had reached the stage 
where units of measurement were necessary, units 
were at first arbitrarily chosen, but it was very early 
recognized that such a course would lead to a multi- 
tude of troublesome factors for reducing to other 



2 NOTES UPON ELECTRICAL MEASUREMENTS. 

units already fixed, and, fortunately, the matter was 
taken up and a consistent system of units devised 
before the arbitrary units had acquired a strong foot- 
hold. 

The system so devised is known as the centimetre- 
gramme-second, or the c.g.s., system of units. In 
order that it may be fully understood, it will be 
necessary to consider briefly the basis upon which the 
system rests. 

It has been said that all physical phenomena are 
the result of matter and motion. Any physical 
measurement may, therefore, be effected by measur- 
ing the mass of matter, and the motion involved. 
But motion implies two elements — space or length, 
and time. The measurement of any physical phe- 
nomenon, therefore, involves the measurement of 
three independent quantities, and for the measure- 
ment of these quantities three independent units 
must be arbitrarily chosen. These are called funda- 
mental units, and from them all other units may be 
derived. 

Fundamental Units. — The fundamental units of 
the c.g.s. system are: for the unit of length, the 
centimetre, which is the one-hundredth part of the 
length of a certain platinum bar deposited in the 
archives of France and declared by government enact- 
ment to be a metre; for the unit of mass the gramme, 
which is the one-thousandth part of a cube of platinum 



THE CCS. SYSTEM OF UNITS. 3 

deposited in the archives of France and declared by 
government enactment to be a kilogramme; for the 
unit of time, the second, which is the 86400th part of 
the mean solar day. 

Characteristics of Matter. — The distinctive char- 
acteristic of matter is its persistence in whatever state 
of rest or motion it may happen to have, and the 
resistance which it offers to any attempt to change 
that state. This property is called inertia. 

The resistance which a body offers to change of state 
is, other things being equal, proportional to its mass. 

Anything that changes the state of a body with 
respect to rest or motion is called force. 

Motion. — Motion may be uniform or varied. Rate 
of motion is called velocity. In a uniform motion, 
velocity is measured by the space passed over in unit 
time. In varied motion, the velocity at any instant 
of time is measured by the space which the body 
would pass over in the next second if, from that 
instant, the velocity were uniform. 

The unit velocity is the velocity of a body which, 
moving with a uniform motion in a straight line, 
describes unit space in unit time. 

Rate of change of velocity is called acceleration. 
Acceleration may be constant or varied. 

When constant, it is measured by the change in 
velocity which occurs in unit time. When varied, its 
value at any instant is measured by the change in 



4 NOTES UPON ELECTRICAL MEASUREMENTS. 

velocity that would occur in unit time if, from that 
instant, the acceleration were constant. 

Unit acceleration is the acceleration of a body 
whose velocity, changing at a constant rate, changes 
by the amount of one unit velocity in the unit time. 

A constant velocity is expressed by the ratio space 
to time : 

»-£ w 

This expression is also true for the instantaneous 
velocity at a given instant of time in varied motion, 
if s represent a space described at that instant in a 
time t so short that during that time the velocity 
may be considered constant. 

In the same way, a constant acceleration is ex- 
pressed by the ratio velocity to time : 

<* = 7> (2) 

v representing the change of velocity in the time /. 
If v and / be taken small enough, formula (2) gives 
the instantaneous acceleration when acceleration is 
variable, just as (1) gives the instantaneous velocity 
in varied motion. 

Uniformly Varied Motion, or Motion in ivliicli 
Acceleration is Constant. — Let v represent the initial 
velocity. Then velocity at end of time / is 

v = v x + at (3) 



THE C.G.S. SYSTEM OF UNITS. 5 

Since the velocities in this case form a series in 
arithmetical progression, the mean velocity is the half- 
sum of the initial and final velocities, or 



v l -f- v v x -f- v x -f- at at 

2 2 ' ' 2 ' 



and the space described in time / is 

s = V -^< (4) 

or 

s = v x t + \at (5) 

Substitute in (4) the value of t from (3): 



* = ^ w 



If the motion start from rest, v x = o, and the 
above formulae become 

v = at ; (7) 

^ = iiy/; (8) 

* = £**'; . . . . . (9) 



s = — (10) 

2a v J 



Force. — Since force is the assumed cause of change 
of motion, it is proper to take the change produced 



6 NOTES UPON ELECTRICAL MEASUREMENTS. 

in unit mass in unit time, that is, the rate of change 
of motion, or acceleration, as a measure of the force 
producing it. But the force required to produce a 
given change is proportional to the mass. Hence the 
force required to produce a given acceleration in any 
mass is proportional to the product of the mass by 
acceleration. 

Since we are free to choose a unit of force, we may 
make the force 

F = ma, (i i) 

m being any mass, and a the acceleration produced in 
that mass by the force F. We may now define the 
c.g.s. unit force as that force which may produce in 
a mass of one gramme the c.g.s. unit acceleration. 
It is called a dyne. 

Work and Energy. — The physical idea of work is 
resistance overcome through space. Assuming a force 
to produce motion in its own direction, work is meas- 
ured by the product of that force by the space through 
which it acts. Hence the following equation: 

W=Fs (12) 

The c.g.s. unit work is the work performed by a 
force of one dyne acting through a space of one cen- 
timetre in the direction of the force. It is called an 
erg. 

When the resistance overcome is inertia, the force 



THE C.G.S. SYSTEM OF UNITS. 7 

overcoming it equals ma\ see (n). If the force con- 
tinue to act until the velocity suffers a change equal 
to v, then the space through which it acts is (10), 

s = — . Substituting these values of F and s in (12), 

W=±mv> (13) 

That is, the work which must be done to impart to the 
mass m the velocity v equals half the product of the 
mass by the square of its velocity. Conversely, if a 
mass m, moving with a velocity v, be brought to rest, 
it will do work represented by the same product. 

Energy is capacity for doing work. Thus the mass 
of the last paragraph has a capacity for doing work 
equal to ^mv\ This is the energy it possesses 
because of its motion. 

A body may also possess energy because of its 
position of advantage 'with respect to some force. The 
energy which a body possesses because of its motion 
is called kinetic energy. That which it possesses 
because of its position is called potential energy. 

Energy is measured in the same units as work. In 
describing the measurement of work it was assumed 
that the displacement of the body was in the line of 
direction of the force, but this may not be the case. 
The body may be constrained to move in a path 
whose direction makes an angle with the line of direc- 
tion of the force. In this case, to obtain the effective 



8 NOTES UPON ELECTRICAL MEASUREMENTS. 

force along the path, the force must be resolved into 
two components, one in, and the other at right angles 
to, the path. Evidently the component lying in the 
path is the only component having any effect to pro- 
duce motion along the path, and the work done is 
measured by the product of this component into the 
space through which the point of application of force 
moves. The component along the path is the projec- 
tion of the force upon the path. If a be the angle 
between the path and the line of direction of the force, 
the effective component along the path is F cos a. 

Practical Units. — The erg is a very small amount 
of work. For practical purposes a larger unit, equal 
to ten million (io T ) ergs, is employed, and is called 
a joule. 

The watt is a unit rate of working. It is work 
performed at the rate of one joule per second. 

Gravitation Units. — Before the adoption of the 
simply related units of the c.g.s. system certain arbi- 
trary units of force and work were in use, and are still 
largely used in engineering practice. Those most in 
use are: a force equal to the weight of a pound of 
matter, called also a pound, and a force equal to the 
weight of a kilogramme at Paris, called also a kilo- 
gramme. It is unfortunate that the name pound 
should have been used for two such totally distinct 
quantities. 

From these units of force are derived the units of 



THE C.G.S. SYSTEM OF UNITS. 9 

work, the foot-pound, being the work done by a force 
of one pound acting through a space of one foot, and 
the kilogramme-metre, being the work done by a force 
of one kilogramme acting through a space of one 
metre; also the units rate of working, the horse-power, 
being work performed at the rate of 550 foot-pounds 
per second, and the chceal-vapeur, work performed at 
the rate of 75 kilogramme-metres per second. 

Heat Units. — Since heat is a form of energy, it is 
important to define here the heat units and their 
relation to the other units of work and energy. 

The British thermal unit is the heat required to 
raise the temperature of one pound of water from 32 
to 33 Fahrenheit. 

The pound-degree centigrade is the heat required to 
raise the temperature of one pound of water from zero 
to i° centigrade. 

The calorie is the heat required to raise the tem- 
perature of one kilogramme of water from zero to 
i° centigrade. 

The lesser calorie is the heat required to raise the 
temperature of one gramme of water from zero to i c 
centigrade. 

Below are given the relations between these various 
units. 

UNITS OF FORCE. 
Kilogramme = 981000 dynes. 
Pound = 444972 " 



10 NOTES UPON ELECTRICAL MEASUREMENTS. 

UNITS OF WORK OR ENERGY. 

Kilogramme-metre = 9.81 X io 7 ergs. 

" = 9.81 joules. 

Foot-pound = 1.356 X 1 o 7 ergs. 

11 = 1.356 joules. 

UNITS RATE OF WORKING. 

Cheval-vapeur = 736 watts. 

Horse-power = 746 " 

UNITS OF HEAT. 

Calorie = 426 kgm. -metres. 

11 = 4160 joules. 

British thermal unit = 778 foot-pounds. 

11 " " = 1055 joules. 

Pound-degree centigrade = 1400 foot-pounds. 

= 1898 joules. 
Joule = .00024 calorie 

= .000948 B. T. U. 

= .000527 pound-deg. C. 

PROBLEMS. 

(i) A body is projected upward with a velocity of 
5000 cm. How high will it rise ? 

(2) A rifle-barrel is 75 cm. long. A rifle-ball of 15 
grms. leaves the rifle with a velocity of 60000 cm. per 
second. Assuming a uniform acceleration, for how 



THE C.G.S. SYSTEM OF UNITS. II 

long a time was the bullet in the rifle ? What was 
the force of the powder in dynes ? in pounds ? 

(3) If the bullet of the last example were stopped 
in a space of 10 cm. by a uniform resistance, what is 
that resistance ? 

(4) What energy— ergs, joules, foot-pounds — did 
the bullet possess on leaving the rifle ? 

(5) How much heat — calories, B.T.U. — can be 
generated in one hour by one horse-power ? 



12 NOTES UPON ELECTRICAL MEASUREMENTS, 



THE MAGNETIC FIELD. 

Definition of a magnet. 

Natural and artificial magnets. 

Bar magnets, horseshoe magnets. 

Distribution of magnetic force. 

Ends where force is manifested called poles. 

The two poles are not alike. 

Any magnet suspended so as to be free to swing in 
a horizontal plane settles with one pole toward the 
north and the other toward the south. One pole 
cannot exist without the other. 

Pole pointing north is in English writings called 
the north pole. 

Mutual action of poles. 

Force extends all around the magnet and to a great 
distance. 

Space around a magnet where its forces are mani- 
fested is called the magnetic field. 

Direction and intensity of forces in different parts 
of the field vary greatly. 

A curved line so drawn in the field that at each 



THE MAGNETIC FIELD. I 3 

point of this line the line of direction of the magnetic 
force at that point is tangent to it is called a line 
of force. 

The direction of the force along the line is assumed 
to be the direction in which a free north pole would 
move in obedience to that force. 

Lines of force indicated by iron-filings and by a 
small needle. 

Lines of force can never cross each other, for, if 
they did, that would mean two directions of the mag- 
netic force at the point of crossing. This is impossi- 
ble. 

Strength of Magnetic Poles. — Forces exerted by 
different poles vary greatly. 

Force exerted by the same pole at different dis- 
tances varies greatly. 

Theory indicates and careful measurements demon- 
strate that the magnetic force exerted by one mag- 
netic pole upon another varies inversely as the square 
of the distance. 

The strength of a magnetic pole is assumed, other 
things being equal, to be proportional to the force 
exerted by it. 

Unit Magnetic Pole. — Since the action between 
two magnetic poles is mutual, the force exerted by 
one pole upon another must be proportional to the 
product of the two pole strengths. Hence, as we are 



14 NOTES UPON ELECTRICAL MEASUREMENTS. 

free to choose the unit strength of pole, we may put 



*-? 04) 



The c.g.s. unit strength of pole is, then, a pole of 
such strength that it repels another equal and similar 
pole at a distance of one centimetre with a force of 
one dyne. 

Intensity of Magnetic Field. — It is often con- 
venient to express magnetic forces in terms of the 
intensity of the magnetic field. The intensity of the 
magnetic field at any point is assumed to be propor- 
tional to the force exerted upon a pole of given 
strength placed at that point. Hence we may put 

F=Hp, (15) 

where H is the intensity of the field. 

The c.g.s. unit intensity of field is now a field of 
such intensity that in it unit pole is actuated by a 
force of one dyne. 

Since unit pole is also actuated by a force of one 
dyne at a distance of one centimetre from another 
unit pole, it follows that unit intensity of field is found 
at a distance of one centimetre from the unit pole. 

These definitions have been given as though one 
pole alone could act to produce a magnetic field, but 
this is a condition that cannot be realized in practice 



THE MAGNETIC FIELD. 1 5 

It has already been stated that one pole could not 
exist alone. It is also true that each pole affects to 
a greater or less extent the intensity and direction of 
the lines of force in the magnetic field. 

Graphical Representation of Magnetic Field. — 
Lines of force may not only be used to indicate the 
direction of the forces in the magnetic field, but they 
may be so drawn as to indicate the intensity also. 
For this purpose they are so drawn that the number 
of lines passing through a square centimetre at any 
point is equal to the number of units expressing the 
field intensity at that point. 

It is customary to refer to the intensity of a mag- 
netic field by saying it is a field having so many lines. 

Fields of from iooo to 16000 lines are met with in 
electric generators and motors. 

The Earth's Magnetic Field. — The lines of force 
of the earth's field in this region lie in a vertical plane 
deviating by some 7 to the west of the geographical 
meridian, and are inclined at an angle of some 75 
with the horizon. 

The deviation from the true north is called the 
declination of the magnetic needle, and the deviation 
from the horizontal is called the inclination or dip. 
Since magnetic needles are usually free to swing only 
in a horizontal plane, they are affected only by the 
horizontal component of the earth's magnetic intensity. 
This is called the horizontal intensity of the earth's 



1 6 NOTES UPON ELECTRICAL MEASUREMENTS. 

magnetic field, and is represented by the symbol H. 
If a be the angle of dip, then 

H = total intensity X cos a. 

Measurement of Field Intensity. — Suppose it is 
required to determine the horizontal intensity of the 
earth's magnetic field. A magnetic needle is sus- 
pended by a fibre of silk so as to be free to swing in 
a horizontal plane. If it be deviated slightly from 
the magnetic meridian and then left to itself, it will 
vibrate back and forth in a time which will be less the 
greater the force, and greater the greater the resist- 
ance offered by its inertia. 

The resistance offered by the inertia of a body to a 
force producing rotary motion depends not only upon 
the mass of the body, but also upon the distribution 
of that mass with respect to the axis of rotation; 
that is, upon a factor of the body which is called 
the moment of inertia. This is a factor that may 
be determined by computation for some bodies of 
regular form, but for irregular bodies it must be 
determined by experiment. We will represent it by 
the symbol /. 

Let NS in the figure represent the magnetic me- 
ridian, and A a magnetic needle deflected from the 
meridian plane. A force at each end of the needle, 
equal to Hp, constitutes a couple which causes the 
needle to vibrate. The effect of this couple to pro- 



THE MAGNETIC FIELD. 



»7 



duce rotation is measured by the product of one of 
the forces by the distance between them, which, when 
the needle is at right angles to the meridian, is the 
length of the needle, hence by Hpl, where / is the 
length of the needle. //, represented by the symbol 



1 



HP 



A 



HP* 



Fig. i. 



My is called the magnetic moment of the needle. 
Then it can be shown that the time of a single vibra- 
tion or oscillation is 



= n \l 



i_ 

HM' 



(16) 



If / and / are determined, HM may be found from 
formula (16). 

But the quantity sought is H, and the above ex- 
periment gives the product HM. 

To find H it is necessary to make another measure- 
ment to determine the ratio of H to M. 



1 8 NOTES UPON ELECTRICAL MEASUREMENTS. 

Let the magnet A of Fig. I be placed as in Fig. 2, 
at some distance to the east or west of a small needle 



/ 



Fig. 2. 

B whose deflection can be accurately measured. A is 
so placed that its axis prolonged passes through the 
centre of B. Then if M be the magnetic moment of 
A y and M x that of By r being the distance between 
the centres, and the deflection of B produced by Ay 
it can be shown that 



2 MM, cos 



= M l ffsin 0\ 



M 
H 



= \r* tan 0. 



(17) 



From (16) and (17) both //'and M may be found. 

The method here described for determining H is 
similar to that employed for determining the intensity 
of gravity by means of the pendulum. 

Horizontal intensity of the earth's field in New 
York is about .17 c.g.s. units. 



THE MAGNETIC FIELD. 1 9 

It has been suggested to adopt as a practical unit 
of field intensity the intensity of a uniform field in 
which there are io 8 c.g.s. lines per cm.\ This unit 
has been called a gauss. A field of ioooo c.g.s. 
units would be T V milligauss. 

PROBLEMS. 

(6) A magnetic needle vibrates 15 times per minute 
in the earth's magnetic field in New York. The 
same needle vibrates 16 times per minute in another 
location. What is the value of H in the second loca- 
tion ? 

(7) Two needles identical in dimensions and mass 
make, at the same place, one 50, the other 30 vibra- 
tions per minute. How do their strengths compare ? 

(8) If two needles known to be of the same strength 
make, at the same place, one 50, the other 30 vibra- 
tions, how is this accounted for ? Explain the relation 
between the two. 

(9) The needle A of Fig. 2 causes B to be deflected 
1 5 . Another needle substituted for A causes a 
deflection of 20 . Required the relative strengths of 
the two needles. 

(10) If in Fig. 2 the distance AB is 50 cm. and the 
needle i? is deflected 15 , what is the relation between 
the magnetic moment of A and the field in which B 
is placed ? 



20 NOTES UPON ELECTRICAL MEASUREMENTS, 



THE ELECTRIC CURRENT. 

When the two terminals of an electric generator are 
joined by a conductor, something takes place which 
is called an electric flow. 

The conductor is said to carry an electric current. 
This current is known only by its effects. 

It heats the conductor. 

It affects a magnetic needle near which the con- 
ductor may be placed. This shows that the current 
develops a magnetic field. 

Field due to Current. — Direction. — Studying the 
effect of the current upon a needle, it is seen that the 
lines of force are concentric circles of which the wire 
is the axis. 

Can be shown by means of iron-filings sprinkled on 
a glass plate. 

The direction of the force in these lines may be 
determined from the following rule: Suppose the air- 
rent flowing from yon. The lines then have the direc- 
tion of the movement of the liands of a watch whose 
dial faces you. 

Intensity. — Let AB y Fig. 3, represent the direction 
of the horizontal component of the earth's magnetism. 



THE ELECTRIC CURRENT. 



21 



Let a conductor carrying a current be placed parallel 
to AB and over or under the magnetic needle NS. 

B 




The lines of force due to the current are at right 
angles to AB, and forces in opposite directions, consti- 
tuting a couple, will act upon the opposite poles of 
the needle. 

These will cause a deflection which is opposed by 
the force of the earth's magnetism, and it is evident 
that the needle will be in equilibrium when its direc- 
tion is that of the resultants of the two pairs of forces, 



22 NOTES UPON ELECTRICAL MEASUREMENTS. 

as represented in the figure. The force clue to the 
earth's magnetism acting upon each pole of the 
needle is Hp. That due to the current is H x p at 
right angles to Hp. If a be the angle of deflection, 
it is plain from the figure that 



or 



H x p — Hp tan a, 
H x = H tan a. . 



(18) 



If the conductor, instead of remaining in the mag- 
netic meridian, is turned with the needle and kept 
parallel to it, the deflecting force will always be at 




Fig. 4. 

right angles to the needle as in Fig. 4, where only 
one force of each pair is shown. Here we have 

H x p — Hp sin a, 



THE ELECTRIC CURRENT. 23 

or 

H x — H sin a (19) 

It may be assumed as self-evident that if all parts 
of a conductor were at equal distances from a needle, 
the force exerted would be proportional to the length 
of the conductor. 

It is shown experimentally that, other things being 
equal, the force exerted by a current upon a needle 
is proportional to the inverse square of the distance. 

It is assumed that, other things being equal, 
strength of current is proportional to the force it 
exerts upon a magnetic pole. Hence we may put 

"=^ w 

where C is the strength of current, L its length, d its 
distance from the pole whose strength is p. 

We may now define the c.g.s. unit current as a 
current of such strength that, flowing in a conductor 
1 cm. long, bent into an arc of a circle of 1 cm. 
radius, it will exert upon the c.g.s. unit magnetic pole 
at the centre of the circle a force of one dyne. 

The c.g.s. unit current has no name. The unit 
used in practice is one tenth of the c.g.s. unit, and is 
called the ampere. 

It is impossible practically to realize the conditions 
stated in the definition of the unit current, because 



24 NOTES UPON ELECTRICAL MEASUREMENTS. 

the conductors by which the current is brought to 
and carried away from the arc of unit length would 
themselves have some influence upon the needle. 
But if the conductor be bent into a complete circle 
of unit radius, the conductors leading to and from the 
circle may then be twisted together so that the cur- 
rents flowing in them in opposite directions neutralize 
each other, and the effect produced is that of the 
circle alone. The length of such a circle being 2 7r, 
the field produced at the centre by unit current flow- 
ing in the conductor will be 2n. 

A conductor forming a circle of radius r will, there- 
fore, produce at its centre a magnetic field : 

tt _C2nr _2n 
Jri l — 5 — — — C. 

r r 

Tangent Galvanometer. — If the plane of such a 
circle coincide with the plane of the earth's magnetic 
meridian, and a short magnetic needle free to swing 
in a horizontal plane be placed at its centre, it will be 
in equilibrium (see (18)), when 

27T 

— C — H tan a (21) 

Hence 6 = tan a (22) 

If the conductor make n turns, 

Hr 

C — tan a. . . . . (23) 

2nn v 3J 



THE ELECTRIC CURRENT. 2$ 

Note that the result is independent of the strength 
of the needle. 

An instrument constructed to realize these condi- 
tions is called a tangent galvanometer because the 
current is proportional to the tangent of the angle of 
deflection. 

Hr 
The quantity is the constant of the galvan- 

ometer. 

Disadvantages of this form of tangent galvanom- 
eter. 

Helmholtz *s Form of Tangent Galvanometer. — In 
this instrument there are two equal coils placed at a 
distance apart equal to their radius, and the needle is 
on the common axis midway between them. 

The constant of this instrument is 



,^t (24) 

11 v ^ J 



Current-measuring Instruments employing Arti- 
ficial Fields. — The tangent galvanometer, depending 
for its indications upon the intensity of the earth's 
magnetic field, can only be used where this intensity 
is constant, or at least free from local disturbances. 
Furthermore, large currents cannot be accurately 
measured by this instrument because the fields pro- 
duced by such currents are very large in comparison 
with the field of the earth. Instruments are therefore 



26 NOTES UPON ELECTRICAL MEASUREMENTS. 

constructed in which a strong artificial field produced 
by a permanent magnet is employed to direct the 
needle. Such instruments must be calibrated by 
direct comparison with some standard instrument. 

Other Current-measuring Instruments. — Instead 
of employing a magnetic field as the directive force to 
oppose the force of the current, a spring may be 
employed. 

Instead of making the magnetic needle movable, 
this may be fixed and the coil made the movable 
element. 

Current-measuring instruments are often called 
amperemeters or ammeters. 

Mutual Action of Currents. — Since every electric 
current produces a magnetic field, there must be a 
mutual action between any two currents placed near 
each other. The following rules apply to this mutual 
action: Parallel currents flowing in the same direction 
attract, flowing in opposite directions they repel. 

Currents making an angle tend to become parallel 
and to flow in the same direction. 

The following general law applies to all cases: 

A current in any magnetic field tends to move in 
such a way as to increase the number of lines of force 
enclosing it. 

Currents may be measured by means of their 
mutual action. For this purpose two coils are so 
connected that the same current traverses both and 



THE ELECTRIC CURRENT. 2J 

the force due to their mutual action is measured. 
Evidently this force is proportional to the square of 
the current. 

Electrodynamometers. 

Electric balances. 

Electrical Quantity. — It is sometimes necessary to 
consider the total quantity of electricity employed 
during a period of time. It is assumed that the unit 
current conveys the unit quantity per second. Hence 
the quantity conveyed in t seconds is 

Q=ct (25) 

The practical unit quantity is the coulomb, which is 
the quantity conveyed by the ampere in one second. 

PROBLEMS. 

(11) Where H=.iy f what is the constant of a 
tangent galvanometer having a coil of fifty turns 50 
cm. diameter ? 

(12) If the needle of such an instrument is deflected 
30 , what is the current in amperes ? What quantity 
of electricity per hour is conveyed by such a current ? 

(13) If it be required to measure a current approxi- 
mating 100 amperes, how large a coil of one turn 
would be required, the allowable deflection being 
6o° ? 

(14) What field intensity exists at the centre of a 



28 NOTES UPON ELECTRICAL MEASUREMENTS. 

circular coil of one turn one metre in diameter, carry- 
ing iooo amperes ? 

(15) A tangent galvanometer has two coils, one 80 
cm. diameter, ten turns, the other 100 cm. diameter, 
eight turns. Currents measured by the two coils re- 
spectively produce the same deflection. What is the 
relation between those currents ? 



POTENTIAL AND ELECTROMOTIVE FORCE. 29 



POTENTIAL 



AND 



ELECTROMOTIVE FORCE. 

Potential is a concept that was introduced into 
mechanics for the purpose of simplifying the study of 
the effects produced in a field of force. The charac- 
teristics of a field are known when we know the 
direction and intensity of the forces exerted at various 
points in it. 

It must be remembered that no force exists in a 
field except when there is present in it some body or 
agent peculiar to that field. For example, in a gravi- 
tation field no force exists except where matter is 
present. In a magnetic field no force exists except 
\ where a magnetic pole is present. The intensity of a 
field at any point is measured by the force which 
would act upon a test unit of the kind to which the 
force is due, if such a unit were present. The actual 
force exerted is the product of the field intensity by 
the number of such units present. 

Now in studying the effects of such forces where 
their directions and intensities vary from point to 



30 NOTES UPON ELECTRICAL MEASUREMENTS. 

point within the field, the problem becomes very 
complicated if we attempt to solve it by taking into 
account the forces themselves. 

But every movement in a field of a body acted 
upon by the forces involves work, and problems 
relating to the effects of such movements are much 
simplified if we express the characteristics of the field 
in terms of the work done in moving a body from one 
point to another in it, instead of in terms of the 
direction and intensity of the forces. 

Difference of Potential. — We define the difference 
of potential between two points in a field of force to 
be a difference of condition between the two points 
which is measured by the work which would be done 
by the forces in the field in moving a test unit from 
one point to the other. The work done in this case 
is independent of the path over which the body is 
m'oved. For it is self-evident that if work is done 
by the forces in moving a body from the point A to 
the point B in a field of force, exactly the same work 
must be done against the forces in moving the body 
back by the same path from B to A. Now if more 
work can be done by the forces of the field by moving 
the body over one path than by moving it over 
another, the body might be made to move from A to 
B by the path giving the greater work, and back to A 
by the path requiring the lesser work, and so work 
could be continually done by simply allowing a body 



POTENTIAL AND ELECTROMOTIVE FORCE. 3 1 

to go from one point to another by one path, and 
back to the first point by another. But this is in- 
consistent with the principle of the conservation of 
energy. 

If Fand V l represent the potentials at the points A 
and B respectively, and s the distance between them, 
it is plain that the average force that is exerted upon 
a test unit as it moves from one point to the other is 

V— V, 
F =z \ and if the points are taken very near 

together F will be the force at the middle point 

V— V x 
between them. But - 1 is the rate of cliange of 

potential with respect to space. Hence the intensity 
of the field at any point is the rate of fall of potential 
at that point. 

Electrical Difference of Potential.— An electrically 
charged body produces a field of force, as indicated 
by the action upon light bodies. 

Two kinds of electrical charges. One charge can 
never exist without the other. 

Bodies having like charges repel, having unlike 
charges attract. 

Charges communicated by contact. 

A light body vibrates between two bodies oppo- 
sitely charged. It is evident that work is doyie by 
the electrical forces producing this vibration. 

Measurement of Potential Difference. — The dif- 



32 NOTES UPON ELECTRICAL MEASUREMENTS. 

ference of electrical potential between two charged 
bodies is equal to the work done in carrying a small 
body charged with the unit quantity of electricity from 
one to the other, and the work done in carrying any 
charged body from one point to another is measured 
by the product of the quantity of electricity upon the 
body carried and the difference of potential between 
the two points. 

If two charged bodies be connected by a metallic 
wire, they will, unless they are connected with some 
electric generator, be discharged or brought to equili- 
brium. Electricity is said to flow from one to the 
other. This constitutes an electric current. If the 
charges upon the bodies are maintained by connecting 
them with some electric generator, a continuous cur- 
rent flows through the wire. In this case, as in the 
case of the vibrating body, the work done is measured 
by the product of the quantity of electricity carried 
by the wire, and the potential difference between the 
two bodies, or 

W=EQ r . % . . . . (26) 

where W is the work done, and E the potential 
difference. 

The c.g.s. unit potential difference may now be 
defined as that potential difference which, in trans- 
ferring the c.g.s. unit quantity of electricity, performs 



POTENTIAL AND ELECTROMOTIVE FORCE. 33 

work equal to one erg. The practical unit potential 
difference is the volt = zo 8 c.g.s. units. 

The rate of working ox power expended is 

p _W_EQ 
^~ t ~ t ' 

. Q - 

or, since - = C , 

P=EC {27) 

Since the volt is 10 s c.g.s. units and the ampere is 
io* 1 c.g.s. units, one ampere flowing with a fall of 
potential of one volt gives IO 7 ergs per second, or 
one watt. 

The test of electrical difference of potential is the 
attraction or repulsion of light bodies, or, in the case 
of charged bodies, the current produced when the 
bodies are connected by a conductor. 

Gold-leaf electrometer. 

Quadrant electrometer. 

Electromotive Force may be defined as anything 
that produces or tends to produce an electrical flow. 
Under this definition, potential difference is an electro- 
motive force, but electromotive force, E.M.F., is a 
much broader term than difference of potential. It 
includes all agencies that tend to produce an electric 
flow. The term electromotive force is usually applied 
to such agencies as tend to disturb the electrical 
equilibrium and bring about difference of potential. 



34 MOTHS UTON KLKC7RTCAL MEASUREMENTS. 

When a glass rod is rubbed with silk an electromotive 
force is developed which transfers electricity from the 
silk to the rod and develops a difference of potential. 

Difference of potential is ahvays due to a disturb- 
ance of electrical equilibrium by an electromotive 
force. 

Electricity, like water, seeks its own level, and left 
to itself, differences of potential would sooner or later 
disappear. 

E.M.F. is measured by the difference of potential 
it can produce. 

Examples of E.M.F. — When copper and zinc 
plates are immersed in dilute sulphuric acid the 
copper becomes positive and the zinc negative. An 
E.M.F. exists, tending to carry electricity across the 
liquid from the zinc to the copper. This E.M.F. is 
independent of the size of the plates. 

When the junction of two metals is heated, an 
E.M.F. is in general developed which carries elec- 
tricity from one metal to the other, producing a 
difference of potential between them. 

When a conductor is moved across a magnetic 
field, an E.M.F. in general exists, causing a transfer 
of electricity from one end toward the other, so 
developing a difference of potential between the two 
ends. 



POTENTIAL AND ELECTROMOTIVE FORCE. 35 

PROBLEMS. 

(16) A potential difference of 200 volts exists 
between two bodies 10 cm. apart. What is the 
mean force acting upon a small body charged with 
one coulomb to carry it across from one body to the 
other ? 

(17) A 16-candle incandescent lamp consumes 
about .5 ampere at 110 volts. How many watts? 
How many horse-power to operate 500 such lamps ? 

(18) How much heat may be developed by a cur= 
rent of 6 amperes at 240 volts ? 

(19) What quantity of electricity is conveyed in 
one hour by a current of 10 amperes ? 

(20) A body charged with 50 coulombs is dis- 
charged in 10 seconds. What is the mean current ? 



36 NOTES UPON ELECTRICAL MEASUREMENTS. 



RESISTANCE AND OHM'S LAW. 

There is no such thing as a perfect conductor of 
electricity. 

That the best conductors offer resistance to the 
flow of electricity in them is shown by the fact 
that whenever a current flows there is always a fall of 
potential along the conductor. 

Resistance is proportional to the length of the con- 
ductor and inversely proportional to its cross-section. 

Specific Resistance. — Different materials forming 
conductors of same length and cross-section vary 
greatly in resistance. The specific resistance of a 
substance may be defined as the ratios of the resist- 
ance of a conductor of that substance to the resist- 
ance of a conductor of same length and cross-section 
of some other substance, taken as a standard. 

Or, the absolute specific resistance of a substance 
is the resistance of a centimetre cube of that sub- 
stance taken between opposite faces. 

Copper and silver have the least specific resistance. 
Other metals have varying specific resistances. Iron 
has about six times and mercury about sixty times 
the specific resistance of silver. 



RESISTANCE AND OHM'S LA W. 37 

Liquids have much higher resistances. The resist- 
ance of the liquids used in galvanic batteries is from 
one to ten million times that of copper. 

Ohm's Law states that the current flowing in a 
conductor is directly proportional to the potential 
difference, and inversely proportional to the resist- 
ance, or, since we have yet to choose a unit of resist- 
ance, we may put 

c=| (28) 

Unit Resistance. — The c.g.s. unit resistance may 
now be defined as that resistance through which the 
c.g.s. difference of potential will carry the c.g.s. unit 
current. 

The practical unit resistance is the ohm = io 9 

c.g.s. units. 

volts 

Hence amperes = -= . 

r ohms 

The megohm — a million ohms, and the microhm 
= one millionth ohm, are units often used. 

Conductivity measures the capacity of a conduc- 
tor to carry current. It is the reciprocal of resistance. 

Insulation Resistance. — No substance is a perfect 
insulator. Gutta percha, one of the best insulators, 
has a resistance 85 X io 19 times the resistance of 
copper. 

The insulation of telegraph lines and cables, and of 



38 NOTES UPON ELECTRICAL MEASUREMENTS. 

insulated wires generally, is given in megohms per 
mile. 

Value of the Ohm. — One ohm is equal to the re- 
sistance of a column of mercury one millimetre in 
cross-section and 106.3 cm - l° n g- 

It is roughly the resistance of a copper wire -fa inch 
in diameter and 50 yards in length. 

Power Consumed by Resistance. — Combining 
equations (27) and (28), we have 

P=CR; (29) 



E' 



This means that in a given conductor the electrical 
power consumed is proportional to the square of the 
current, or to the square of the fall of potential along 
that conductor. Since the electrical energy expended 
in a conductor develops heat, the heating effect of a 
current in a given conductor is proportional to the 
square of the current. 

Divided Circuits. — When the terminals of an elec- 
tric generator or any two bodies between which a 
difference of potential is maintained are joined by 
two or more conductors, current flows through each 
of them in accordance with Ohm's law. 

Conductors so connected are said to be joined in 



RESISTANCE AND OHM'S LAW. 



39 



multiple arc, or in parallel. Figs. 5 and 6 are typical 
representations of such an arrangement. 



TERMINAL 




TERMINAL 



generator 
Fig. 5. 



If E be the difference of potential between the two 
bodies to which the conductors are joined, and r,, r /yl 



u 
u 



~U~ 



generator 
Fig. 6. 



r iin etc., are the resistances, respectively, of the 
several conductors, the currents flowing will be 



c, 



c -*• 

r 

c -■£• 

etc. etc. 



40 NOTES UPON ELECTRICAL MEASUREMENTS, 



Evidently the total current is 

E E E . 

r, r tt r 



C =*+V-+ — + etc ' = R> ' • (30 



where R is the equivalent resistance of the several 
conductors. From (31) R can be calculated when 
r jy r y/ , etc., are known. For example, (31) gives for 
three conductors 

whence 

fa r „ + r, r„, + r u r„)R = r, r n r ul ■ 

R = , ' " '" . . . (32) 

When two conductors are connected in multiple, 
the fall of potential is the same along both. Evi- 
dently for every point on one there must be a point 
on the other having the same potential. If two such 
points be connected by a wire, no current will flow 
through that wire. 

In Fig. 7 let ABD y ACD y be two conductors 
joining the terminals of the generator 5. The point 
C on ACD } which has the same potential as B on 
ABDy may be found by connecting B to one terminal 
of a doiicate current-indicator G, to the other ter- 
minal of which a wire is connected which may be slid 
along A CD. When G indicates no current, the point 
C is found. 



RESISTANCE AND OHM'S LAW, 



41 



Let r, r , r /yl r //y , be the resistances of the several 
sections of the conductors as marked on the figure. 




Fig. 7. 

Let e be the fall of potential from A to B and from 
A to C. Let ^ be the fall of potential from B to D 
and from C to Z>. Then, since the same current flows 
through AB and BD, 



e 
r 



r, 



For a similar reason 



e 

T., 



Dividing (34) by (33), 



r 



r 






(33) 



(34) 



(35) 



which shows the relation between the four resistances. 



PROBLEMS. 



(21) Two conductors in multiple arc have resist- 
ances of 24 and 30 ohms respectively. What is the 



42 NOTES UPON ELECTRICAL MEASUREMENTS. 

equivalent resistance ? A current of 8 amperes flows 
in the circuit. What is the potential difference 
between the ends of the conductors ? What current 
flows through each ? 

(22) Four conductors, of 1, 2, 3, 4 ohms respectively, 
are in multiple. What is the equivalent resistance ? 

(23) A galvanometer has a resistance of 4500 ohms; 
it is desired to place in parallel with it a resistance 
that shall shunt away from it T 9 F the current. What 
must be that resistance ? 

(24) What is the equivalent resistance of 500 in- 
candescent lamps in parallel, each lamp having a 
resistance of 200 ohms ? 

(25) What current will the lamps of the last prob- 
lem consume at a potential difference of 110 volts? 
Suppose the leads from the generator to the lamps 
have a resistance of 0.01 ohm. What must be the 
potential difference at the generator to maintain 110 
volts at the lamps ? 

(26) If the incandescent lamps of problem (24) are 
connected five in series and these groups connected 
in multiple, what is the equivalent resistance ? What 
current will be consumed if each lamp carries the 
same as before ? What potential difference between 
the supply leads will be necessary ? 



MEASUREMENTS OF ELECTRICAL QUANTITIES. 43 



PRACTICAL MEASUREMENTS OF ELEC- 
TRICAL QUANTITIES. 

In the preceding lectures the relations between the 
several electrical quantities have been brought out, 
and the bases upon which the several units of meas- 
urement have been defined and established have been 
fully discussed. It is, of course, possible to measure 
these electrical quantities by methods based upon 
those relations and definitions. Measurements by 
such methods are called absolute measurements. 

When treating of electric currents it was shown how 
the value of a current could be determined in abso- 
lute measure by means of the tangent galvanometer. 
The methods for the absolute measurement of poten- 
tial and resistance are, however, too tedious and com- 
plicated to be made use of for general measurements, 
and are only resorted to for the purpose of construct- 
ing standards with which the quantities to be meas- 
ured are thereafter compared. In the measurement 
of currents, even, the absolute methods are unsuitable 
for general work, and standards have been devised 
which render unnecessary the absolute determination 
of any current. After comparing the results of the 



44 NOTES UPON ELECTRICAL MEASUREMENTS, 

most accurate absolute determinations the Electrical 
Congress held in Chicago in 1893 fixed upon the fol- 
lowing as the physical representatives of the electrical 
units: 

"As the Unit of Current, the International Am- 
pere, which is one tenth of the unit current of the 
c.g.s. system of electromagnetic units, and which is 
represented sufficiently well for practical use by the 
unvarying current which when passed through a solu- 
tion of nitrate of silver in water and in accordance 
with the accompanying specification deposits silver at 
the rate of 0.001 1 18 grammes per second." 

Similarly the International Volt is declared to be 
represented sufficiently well " by -ffff of the E.M.F. 
between the poles or electrodes of the voltaic cell 
known as Clark's cell at a temperature of 15 C, and 
prepared in the manner described in the accompany- 
ing specification/' 

And the International Ohm is declared to be 
c< represented by the resistance offered to an unvary- 
ing electric current by a column of mercury at the 
temperature of melting ice, 14.4521 grammes in mass, 
of a constant cross-sectional area, and of the length 
of 106.3 centimetres." The cross-sectional area of 
such a mass of mercury 106.3 cm - in length, is one 
square millimetre. 

But it is not convenient in ordinary measurements 
to make use of these official representatives of the 



MEASUREMENTS OF ELECTRICAL QUANTITIES, 45 

units. For practical use instruments are constructed 
by which the electrical quantities may be measured 
much as we measure length by the foot-rule or tape- 
measure, or mass by the balance and weights. Ordi- 
narily we accept the accuracy of such instruments as 
we accept the accuracy of the weights of the balance, 
upon the reputation of the manufacturer; but it must 
be remembered that the electrical instruments, espe- 
cially those for measurement of potential and current, 
are likely to change with time, and all are liable to 
accidental derangements, and it is not safe to trust to 
their accuracy as we trust to the accuracy of the foot- 
rule or tape-measure for an indefinite period. For 
all important measurements the instruments employed 
should be carefully tested by comparison with stand- 
ards of known accuracy. 

MEASUREMENTS OF RESISTANCE. 

Instruments. — Certified standard resistances. 

Resistance sets. These are sets of resistance coils 
so arranged that any resistance from that of the 
smallest coil to the sum of all the resistances in the 
instrument may be employed at pleasure. There 
are two principal arrangements: coils of 1, 2, 2, 
5, 10, 20, 20, 50, etc., ohms, or ten unit coils, ten 
10-ohm coils, etc., are arranged in series, with pro- 
vision for cutting in or out of circuit any desired 
portion, 



46 NOTES UPON ELECTRICAL MEASUREMENTS. 

Methods of Measurement. — First. By direct com- 
parison with known resistances ; 

(a) By substitution. This consists in noting the 
deflection of a galvanometer when connected in cir- 
cuit with the unknown resistance, then substituting 
for the unknown resistance known adjustable resist- 
ances, and adjusting these until the same deflection is 
obtained. The known resistance is then equal to the 
unknown. 

(b) By the differential galvanometer. The differen- 
tial galvanometer is an instrument having two coils of 
equal resistance, so adjusted as to have exactly the 
same influence upon the needle. With equal currents 
flowing in opposite directions in these two coils the 
needle would be undisturbed. To use the instru- 
ment, the unknown resistance is connected in circuit 
with one coil, and adjustable known resistances in 
circuit with the other, the two circuits being con- 
nected in multiple arc between the terminals of some 
electric source. When the known resistances are so 
adjusted that the galvanometer needle suffers no 
deflection, the known and unknown resistances are 
equal. 

Second. By fall of potential. This is a method 
especially applicable to the measurement of small 
resistances. The resistance to be measured is con- 
nected in circuit with a galvanometer which will 
measure the current flowing. The difference of po- 



MEASUREMENTS OF ELECTRICAL QUANTITIES. 47 

tential between the two ends of the resistance is then 
measured by means of some instrument for measuring 
potential differences. If C be the current and e the 
difference of potential, then, from Ohm's law, 



R = 



C 



The figure below illustrates the arrangement. R is 
the resistance to be measured, G the galvanometer, 
and Fthe potential instrument. 

Third. By Wheatstone s bridge. This is an appa- 
ratus utilizing the principle illustrated in Fig. 7. 




Fig. 8. 

Suppose any one of the resistances of Fig. 7 to be 
unknown; it can be determined from equation (35). 
As the instrument is usually constructed, two of the 
resistances, as r y , r„, have a simple ratio, as 1 : I, 
1 : 10, 1 : 100. A third resistance, r, is known and 
adjustable. The unknown resistance is then con- 



48 NOTES UPON ELECTRICAL MEASUREMENTS. 

nected as r //r The instrument is often called Wheat- 
stone's balance; r s and r /y are called the arms of the 
balance. The manipulation consists in varying the 
resistance r until the needle G is undisturbed by the 
closing of the circuit. 

Measurement of Insulation Resistance. — Insula- 
tion resistances up to ten megohms may be measured 
by means of the Wheatstone's bridge, but for the 
measurement of very high insulation resistances it is 
customary to use a delicate galvanometer which will 
give a deflection equal to one scale division for a 
difference of potential of one volt through a deter- 
mined resistance of several megohms. This galva- 
nometer is merely put in circuit with the insulation 
resistance to be measured, and the deflection for a 
given potential difference noted. 

Measurement of Very Small Resistances. — For 
such measurements methods must be employed by 
which the resistances of the connections and contacts 
by which the resistance to be measured is connected 
to the apparatus are eliminated. The fall of poten- 
tial method illustrated in Fig. 8 permits this. The 
connections a, b to the potential instrument V 
are so made as not to include the contact resist- 
ances by which the battery is connected to the resist- 
ance R. 

The method as before described necessitates the 
accurate observation of current and potential. The 



MEASUREMENTS OF ELECTRICAL QUANTITIES. 49 

arrangement shown in Fig. 9 does away with the 
necessity of observing either of these quantities, 
but permits, instead, the direct comparison of the 
resistance to be measured with a known adjustable 
resistance. A is the unknown resistance, R a known 




Fig. 9. 

resistance, such as a graduated wire. P is a delicate 
galvanometer, the value of whose indications need not 
be known. The operation consists in sliding the con- 
tacts cd along R until P shows the same deflection 
whether connected with R or A. The resistance 
between a and b is then the resistance between c 
and d. 

Measurement of Resistance of Electrolytes. — 
When an electric current flows through an electrolyte 
it not only does work in overcoming the true resist- 
ance, but it also does work in decomposing the elec- 
trolyte. This latter work is done in overcoming what 
is called the counter-electromotive force of the liquid. 
This is an apparent resistance which is independent 



50 NOTES UPON ELECTRICAL MEASUREMENTS. 

of the dimensions of the liquid column, and depends 
only upon the nature of the liquid, assuming the 
liquid to have no action upon the electrodes. Means 
by which this apparent resistance may be eliminated 
must be employed for measuring the true resistance. 



l c 





Fig. io. 



Fig. io shows one method, ab is a U tube contain- 
ing the electrolyte, G is a battery, and R an adjust- 
able resistance; c and d are platinum plates nearly 
filling the tube. Let R be adjusted until the galva- 
nometer gives a convenient deflection. Now let one 
of the platinum plates be lowered a measured dis- 
tance — to e, say. Now adjust R until the galva- 
nometer shows the same deflection as before. The 
increase in R is the resistance of the column of liquid 
ce. 



MEASUREMENTS OF ELECTRICAL QUANTITIES. 5 1 

If a column of an electrolyte is traversed by a 
rapidly alternating current, no permanent decomposi- 
tion occurs, and no work is done except in overcom- 
ing the true resistance. The resistance may then be 
measured by any of the methods before described for 
measuring the resistance of ordinary conductors, but, 
instead of the galvanometer, an electrodynamometer 
or some instrument affected by alternating currents 
must be used. If a Wheatstone's bridge is employed 
with alternating currents for measuring the resistance 
of an electrolyte, a telephone may conveniently be 
used in place of the galvanometer to indicate when 
a balance is obtained. 

The resistance of an electrolyte varies greatly with 
temperature and, if a solution, with the degree of 
concentration. Measurements are of no value unless 
these conditions are noted. 

Resistance of Batteries. — The measurement of 
the resistance of battery-cells presents some difficul- 
ties on account of the electromotive force. The 
following are some of the methods employed: 

(a) Two similar cells are connected by two like 
poles so that their electromotive forces are opposed. 
Their joint resistance may then be measured as in the 
case of ordinary conductors. 

{b) The potential difference between the poles of 
the cell on open circuit is measured. This gives the 
E.M.F. of the cell. Represent it by E. The circuit 



52 NOTES UPON ELECTRICAL MEASUREMENTS. 

of the cell is then closed through a known resistance 
r, and the potential difference between the poles again 
measured. Call this e. If R be the resistance of the 
cell, the current flowing is 

u ~~" R + r~ r* 

rE = eR -f- er ; 

E — e 
R =—J-* (36) 

!{* = $£, R= r. 

Hence we may observe the potential of the cell on 
open circuit, then close the circuit through a resist- 
ance which is so adjusted as to reduce the potential 
to one half. The known resistance in circuit is then 
equal to the resistance of the cell. 

This method assumes that the E.M.F. of the cell 
remains constant during the measurements. This is 
not necessarily true. 

(c) By Mance's method, in which the cell whose re- 
sistance is to be measured is connected as the unknown 
resistance in the Wheatstone's bridge, as in the dia- 
gram, Fig. 1 1. The key K is inserted in place of the 
usual battery. The galvanometer G must be one 
whose deflection will not be too great. It may be 
necessary to put a known resistance in the branch X 
in series with the cell, to obtain a convenient deflec- 
tion. Having chosen a convenient ratio for the arms 



MEASUREMENTS OF ELECTRICAL QUANTITIES. 33 

a and b y R is adjusted until the deflection of the 
galvanometer G is unchanged by opening and closing 
the key K. It can be shown that, when this condi- 




FlG. II. 



tion is satisfied, the usual relation between the resist- 
ances of the bridge branches exists. That is, 



a 

~b 



R_ 

x ' 



From this Xis determined. If a known resistance 
was connected in series with the battery-cell in order 
to obtain a suitable deflection of the galvanometer, 
this must be subtracted from X to give the cell resist- 
ance. 



MEASUREMENT OF CURRENT. 

Measurements of current by means of the tangent 
galvanometer have already been fully explained. 
Other current-measuring instruments have been briefly 



54 NOTES UPON ELECTRICAL MEASUREMENTS. 

described. These are usually adjusted by the manu- 
facturers to read directly in amperes, and it is only 
necessary to connect the instrument in the circuit 
where the current is to be measured, and note the 
reading. It is important in the use of all such instru- 
ments to place them where they will be uninfluenced 
by outside magnetic forces. 

Current Measured by Fall of Potential. — In a 
conductor carrying a current there is always a fall of 
potential between its ends. This fall for the same 
conductor is proportional to the current, hence may 
be taken as a measure of the current. Fig. 12 shows 




nrwr\ 

Fig. 12. 

diagrammatically the arrangement of an instrument 
designed to utilize this method of measuring currents. 
R is the fixed resistance and A an instrument whose 
indications are proportional to the potential differ- 
ences between its terminals. It is not necessary to 
know the resistance R. The instrument may be cali- 
brated by direct comparison with some standard 
current-measuring instrument. 
Examples of such instruments. 



MEASUREMENTS OF ELECTRICAL QUANTITIES. 55 



MEASUREMENT OF POTENTIAL. 

By Electrostatic Forces. — This consists essen- 
tially in the measurement of the electrostatic forces 
existing between two bodies charged to the difference 
of potential to be measured. Instruments designed 
for such measurements may be so constructed that 
the potential difference may be computed from their 
dimensions and the forces observed. This is an 
absolute measurement, but is applicable only to the 
measurement of large potential differences. 

The Quadrant Electrometer. 

The Attracted-disk Electrometer. 

By Comparison with a Known Potential Differ- 
ence. — Fig. 13 illustrates this method. ab is a 



^ 



0i 



Fig. 13. 

graduated resistance, such as a long wire or a series 
of resistance coils. G is a generator capable of main- 



$6 NOTES UPON ELECTRICAL MEASUREMENTS. 

taining a constant potential difference between a and 
b, which must be known. C is a battery-cell whose 
E.M.F. is to be measured, so connected in the 
branch aCd that its E.M.F. is opposed to the E.M.F. 
acting along that branch in consequence of the fall of 
potential along ab. Now the point of contact d is 
moved along the resistance ab until the galvanometer 
/ is undeflected. The potential difference between a 
and d now balances the E.M.F. of the cell C. If E 
be the potential difference between a and b, and E, 
the E.M.F. of the cell C y we have 

res. ab E 



res. ad E x " 

An instrument so constructed as to provide for the 
convenient application of the above method is called 
a potentiometer. 

Different forms of potentiometer. 

By the Current produced in Circuit of Given Re- 
sistance. — It follows from Ohm's law that E.M.F. 
is proportional to the current it develops in a given 
resistance. Hence the readings of any galvanometer 
are proportional to the potential difference between 
its terminals. If a galvanometer of high resistance 
be connected between two points differing in poten- 
tial, its indications will be proportional to that poten- 
tial difference, and the instrument may be graduated 



MEASUREMENTS OF ELECTRICAL QUANTITIES. 5/ 

to give the potential difference directly in volts. It 
is then called a voltmeter. 

Different forms of voltmeter shown and described. 

Since closing a circuit between two points may 
alter the potential difference between those points, 
the use of such voltmeters as are described above is 
inadmissible, except where the generating source is 
such as to maintain that potential difference notwith- 
standing the current consumed by the voltmeter. 

By the Ballistic Galvanometer. — If two bodies, as 
the two coatings of a Leyden jar or other condenser, 
be connected to two points of an electric circuit, those 
bodies will be charged to whatever potential differ- 
ence may exist between those points. If they are 
afterward connected through a suitable galvanometer, 
they will be discharged, and the discharge-current 
flowing through the galvanometer coil will give the 
needle a sudden impulse, causing it to swing off a 
certain distance depending upon the quantity of elec- 
tricity discharged, and this again is proportional to 
the potential difference and to the capacity of the 
condenser receiving the charge. 

In this way potential differences may be measured 
by means of the galvanometer without actually form- 
ing a circuit and taking current from the source. 

A galvanometer suitable for this purpose is one 

whose needle swings freely, with the smallest possible 

I retarding influence, so that its needle once disturbed 



58 NOTES UPON ELECTRICAL MEASUREMENTS. 

will swing for a long time before coming to rest. It 
should also have a slow rate of vibration. Such an 
instrument is called a ballistic galvanometer. 

TESTING AND CALIBRATING INSTRUMENTS. 

Electrical instruments are liable to derangement 
and must be frequently tested to determine whether 
changes have occurred. This is especially true of 
instruments for the measurement of current and 
potential. Where a high degree of accuracy is re- 
quired, instruments must be compared with accredited 
standards, their errors determined, and the necessary 
corrections applied when the instruments are used for 
measurements. 

Tests of Resistance Sets. — It is seldom required 
to reproduce the mercury standard ohm for purposes 
of comparison. A certified standard one-ohm coil 
may be used instead. The one-ohm coil of the set is 
compared with the standard. Then this one-ohm 
coil plus the standard is compared with the two-ohm 
coil of the set, and so on until all are compared. In 
a decade set each one-ohm coil would be compared 
with the standard, then the ten one-ohm coils in 
series with each ten-ohm coil, etc. 

Method of Comparison. — A form of Wheatstone's 
bridge known as the divided-metre bridge is usually 
employed for comparing resistances. Fig. 14 shows 
the arrangement, AB is a wire of as nearly as possi- 



MEASUREMENTS OF ELECTRICAL QUANTITIES. 59 

ble uniform cross-section. G, G iy etc., are heavy 
copper strips. E and F are two nearly equal resist- 
ances, which, to give the greatest sensibility, should 
be about the value of the resistances to be compared, 
which are connected in the two gaps C, D. lis now 
moved along the wire AB until the galvanometer is 
undeflected by closing the circuit. The resistances 
at C and D are now interchanged. It is plain that, 
if they are equal, the galvanometer will still be un- 



ii- 



c 



O m( 



G, 



(JT| 1 j) ~6] \WT 



D 

12] [H 
G,„, 







Fig. 14. 



deflected by closing the circuit. If C and D are not 
equal, / will have to be moved in order to obtain a 
balance after the interchange. It is easy to show 
that the resistance of the bridge-wire between the two 
positions of / equals the difference between the 
resistances of the two coils that are being compared. 
The bridge-wire AB in the standard form of this 
apparatus is one metre in length, and under it is a 
scale divided into millimetres; hence the name. If 
the resistance of one millimetre of the bridge-wire is 
known, the difference in fractions of an ohm between 
two coils under comparison is at once given. 



60 NOTES UPON ELECTRICAL MEASUREMENTS. 

Calibration of the Bridge-wire. — It is rarely the 
case that the resistance of the bridge-wire is uniform. 
To determine its resistance in different parts of its 
length it must be calibrated. To effect the calibra- 
tion the resistances E and F are removed, and the 
points m and n are connected by a wire as shown by 
the dotted line. The galvanometer connection K is 
now made on the wire ;//// by a movable contact-piece. 
Place at C a standard one-ohm coil, and at D a i.oi- 
ohm standard; then effect a balance by moving either 
/ or K. Now interchange C and D and balance again 
by moving /. Evidently the resistance of the wire 
between the two positions of /is .01 ohm. Return 
C and D to their former positions, and, without mov- 
ing /, balance by sliding K along mn. Again inter- 
change C and D and balance by moving I. This will 
give another .01 -ohm section of AB. Continuing 
this process step by step, the whole wire may be sub- 
divided into .01 -ohm sections, from which the resist- 
ance between any two points may be obtained. 

In making such tests as those described, arrange- 
ments must be made for maintaining a constant tem- 
perature, and the connections at C and D must be so 
made that their resistances are extremely small and 
are invariable during all the interchanges. 

Tests of Current Instruments. — First. Compari- 
son with a standard. The instrument is connected in 
circuit with a silver voltameter and the test conducted 



MEASUREMENTS OF ELECTRICAL QUANTITIES. 6l 

as directed on page 106 of " Physical Units" by 
Professor Magnus Maclean. This gives the true 
value of the reading for one point of the scale. 

Second. Comparison of different scale-readings. 
Having found the value of the scale-reading for one 
point, the instrument is connected in circuit with a 
variable resistance whose values can be accurately 
determined, and a source of electricity giving a con- 
stant difference of potential. The constancy of the 
potential may be determined by the use of any 
potential indicator. It is not necessary that its value 
should be known. The test consists in varying the 
current by means of the variable resistance, and 
noting the resistance and corresponding readings of 
the instrument. 

Tests of Instruments for Measuring Potential. 
— Voltmeters may be tested by means of the poten- 
tiometer and the standard Clark cell. Referring to 
Fig. 12, ab is the potentiometer resistance, which 
should be as much as 10,000 ohms. V is the volt- 
meter to be tested, and C the standard cell. The 
generator G should have an E.M.F. greater than the 
highest reading of the voltmeter, and it will be con- 
venient if its E.M.F. is adjustable. If this is not the 
case, the potential difference between a and b may be 
varied by shunting a part of the current away from 
ab. 

The test consists in varying the potential difference 



62 NOTES UPON ELECTRICAL MEASUREMENTS. 

between a and b until the voltmeter gives a conven- 
ient reading. Then move the connection d until /is 
undeflected. We then have 

value of volt- ) res. ab „ „ _, „ —. . 

,. V = j X E.M.F. of Clark cell. 

meter reading ) res. ad 

The E.M.F. of a Clark cell for any temperature / 

is 

i.434[i - o.ooo77(/- 15)]. 

Clark cells are themselves liable to derangement. 
It is best to have several of them, and compare them 
with each other by means of the potentiometer. If 
two or three of them agree at a given temperature, it 
is safe to assume that their E.M.F. at 15 C. is 1.434 
volts. 

When using a Clark cell it is necessary to place in 
series with it a protecting resistance of several thou- 
sand ohms to prevent passing through it too large a 
current. This resistance may be shunted for the pur- 
pose of obtaining a higher sensitiveness when the 
potentials are nearly balanced. 



HEATING EFFECTS OF THE CURRENT. 63 



HEATING EFFECTS OF THE CURRENT. 

When a current flows through a homogeneous con- 
ductor, that conductor is always heated. 

Equation (29) shows that the energy spent in such 
a conductor per second is C'R — directly proportional 
to the resistance, and to the square of the current. 
C being given in amperes and R in ohms, the heat gen- 
erated is .00024CV? calories per second = .00094SCV? 
British thermal units per second. 

A 16-candle incandescent lamp consumes about 0.5 
ampere at 110 volts. Its resistance is, therefore, 220 
ohms. The energy expended in it is I IO X .5 = 220 
X -5 2 =" 5 5 watts = 55 X .00024 = -0132 calorie per 
second = .074 H. P. 

Temperature of a Conductor carrying Current. 
— The temperature to which a conductor is raised 
by the current depends not only upon the rate of 
development of heat in it, but upon the rate at which 
heat escapes. The temperature becomes permanent 
when heat escapes as fast as it is generated. 

For a round wire of given length the resistance is 
proportional to the inverse square of the diameter, 
while the radiating surface is directly proportional to 



64 NOTES UPON ELECTRICAL MEASUREMENTS. 

the first power of the diameter. The heat generated 
by the same current in a wire whose diameter is d t as 
compared to that generated in a wire whose diameter 

is unity, is, therefore, -j- a , while the surface from which 

d 

heat escapes is — times as great. The heat which 

must escape from unit surface of the larger wire is, 
therefore, -j- 3 of that which must escape from unit 

surface of the smaller wire. If heat escaping per unit 
surface is proportional to the difference of tempera- 
ture between the wire and its surroundings, the larger 

wire will rise in temperature only --=-, as much as the 

smaller. 

Equation (30) shows that the heat developed in a 

wire is proportional to -75-. 

This means that for the same difference of potential 
the heat generated in conductors of different resist- 
ances is inversely proportional to the resistance, and, 
since resistance for wires of the same length is pro- 
portional to the inverse square of the diameter, the 
heat generated in such wires when subjected to the 
same potential difference is proportional to the square 
of the diameter directly. Hence if two wires, diam- 
eter unity and diameter d> are connected in multiple 
between the terminals of an electric generator, d 



HEATING EFFECTS OF THE CURRENT. 65 

times as much heat will be generated in the wire 
whose diameter is d. But its surface is only ci times 
as great, hence d times as much heat must escape 
from unit surface, which requires that it rise in tem- 
perature d times as much. It follows that a coarse 
wire will be heated to a higher temperature than a 
fine one of the same length when connected across a 
circuit where the difference of potential is the same. 

INCANDESCENT LIGHTING. 

Description of the incandescent lamp. 

Incandescent lamps are usually connected in multi- 
ple arc across a circuit from a generator capable of 
maintaining a constant difference of potential, whether 
lamps in use are few or many. 

Fig. 6, on page 39, may be taken as a diagrammatic 
illustration of such a system. 

Economy requires High Temperature. — Econ- 
omy in incandescent lighting requires that the carbon 
filament shall be maintained at as high a temperature 
as it will bear without a too-rapid deterioration, for 
the reason that the ratio of light emitted to energy 
consumed increases very rapidly as the temperature 
increases. At the best, only about 5 per cent of the 
energy expended in the lamp appears as light; the 
remaining 95 per cent is dark heat, useless for illumi- 
nation, and therefore wasted. 



66 NOTES UPON ELECTRICAL MEASUREMENTS. 

Incandescent Lamps of Different Candle-power. 

— Since potential is constant, the power expended in 
an incandescent lamp is inversely proportional to its 
resistance (see equation (30)). 

Let it be required to construct a 32-candle lamp, 
that is, a lamp of twice the usual illuminating power. 
This requires that twice the power be expended; 
hence that the resistance be reduced to one half. 
Since the temperature must remain constant, the 
radiating surface must be doubled. 

Let / be the length, b the width, and d the thick- 
ness of the 16-c.p. filament, and let / l b 1 d l be the same 
for the 32-c.p. filament. Resistance is proportional 

to -7-7; hence 
ba 

Vd = 2 h {a) 

Radiating surface is proportional to l(b -j- d); hence 

2 /(b + d) = / 1 (b 1 + d l ) (b) 

In these two equations there are three unknown 
quantities. The values are therefore indeterminate; 
but if we assume the value of one, the other two are 
fixed. Assume, for example, /, = /; then from (a) 

b x d x = 2bd\ 
from (b) 

b i + d x = 2(b+d). 



HEATING EFFECTS OF THE CURRENT. 6? 

That is, we must double the cross-section and also 
double the periphery. To do this requires that the 
ratio of width to thickness must be changed, as will 
be seen by finding the values of b y and d x from the 
two equations above. 

ARC LIGHTING. 

Description of arc lamp. 

Light mainly proceeds from a small intensely 
heated surface of the positive carbon, called the 
crater, from which the arc springs. Some comes from 
the red-hot ends of carbons. Very little comes from 
the arc itself. 

Source of the Light. — In almost all arc lamps used 
for illumination the upper carbon is made positive 
and the crater is a small concave surface at its end. 
Light can only issue from this in a direction obliquely 
downward. Hence an arc lamp gives a very un- 
equally distributed illumination. A so-called 2000- 
candle arc lamp gives its most intense light — about 
1500 to 2000 candles—at an angle of 45 to 6o° below 
the horizontal. From this direction the intensity 
rapidly diminishes until, on the horizontal plane 
through the lamp, the illumination amounts to only 
200 to 300 candles. 

As a result, a lamp suspended at a height of, say, 25 
feet gives a most intense illumination on the giound 



63 NOTES UPON ELECTRICAL MEASUREMENl'S. 

below over a circle of about 50 feet in diameter, 
beyond which the illumination rapidly diminishes. 

The so-called 2000-candle arc lamp consumes about 
10 amperes at 45 volts, or 450 watts. Of this 45 volts, 
about 39 volts seems to be expended on the surface 
of the crater, the remainder being expended in over- 
coming the resistance of the arc. That is, about 
seven eighths of the power is expended in the crater. 

About 10 per cent of the power expended appears 
as light, the remaining 90 per cent appearing as dark 
heat. 

The light produced by an arc lamp depends almost 
wholly upon the current, very little upon the poten- 
tial difference so long as this is above 40 volts. Upon 
the latter depends the length of arc or the amount 
of separation of the carbons. About 45 volts is 
required to maintain a separation that will permit the 
free emergence of the light from the crater. Beyond 
this very little is gained by increase of potential. 

Arc lamps are usually connected in circuit in scries; 
hence the same current flows through all of them. 
The E.M.F. required for the operation of such a cir- 
cuit is the E.M.F. required for one lamp multiplied 
by the number of lamps. 

Enclosed arc lamps. 



HEATING EFFECTS OF THE CURRENT. 69 

PROBLEMS. 

(27) If a round filament 4 inches long, .012 inch 
diameter, is suitable for a 16-c.p. lamp, what are the 
dimensions of a similar filament suitable for 32 c.p.? 

(28) Fifty arc lamps, requiring a current of 10 am- 
peres at 45 volts each, are connected in series in a 
circuit 5 miles long, the conducting wire having a re- 
sistance of 0.25 ohm per thousand feet. What must 
be the E.M.F. of the generator ? 

(29) What energy (watts, horse-power) is consumed 
in the lamps of the last problem ? What in the cir- 
cuit ? 

(30) A 10-ampere, 45-volt arc lamp is connected in 
parallel with incandescent lamps across a no-volt 
circuit. What resistance must be used in series with 
the lamp to reduce the potential to 45 volts ? How 
much energy is expended ? How much in the lamp ? 

(31) If two lamps like that of the last problem are 
placed in series across a 110-volt circuit, what resist- 
ance is required in series with them ? How much 
energy is consumed ? How much is wasted ? 






70 NOTES UPON ELECTRICAL MEASUREMENTS, 



CHEMICAL EFFECTS. 

Decomposition of Salts. — When two platinum 
plates immersed in dilute sulphuric acid are connected 
to the terminals of an electric generator, oxygen gas 
escapes in bubbles from the positive plate, and hy- 
drogen from the negative plate. The relative pro- 
portions of oxygen and hydrogen are exactly the pro- 
portions in which they unite to form water. 

In this experiment water is said to be decomposed 
by the current. 

If the same plates be immersed in a solution of 
copper sulphate, copper is deposited on the negative 
plate, while oxygen escapes from the positive plate. 

The operation here evidently is that the current 
separates the CuS0 4 into Cu and S0 4 , the Cu appear- 
ing at the negative and the S0 4 at the positive 
plate. But S0 4 attacks and decomposes the water, 
liberating oxygen and forming sulphuric acid: 

S0 4 +H 2 = H 2 S0 4 + 0. 

If a copper plate be substituted for the positive 
platinum plate, oxygen no longer escapes, but the 



CHEMICAL EFFECTS. 7 1 

S0 4 combines with the copper, forming CuS0 4 , so 
maintaining the strength of the solution. 

If the platinum plates are immersed in solution of 
sodium sulphate, Na 2 S0 4 , hydrogen escapes from the 
negative and oxygen from the positive plate, while 
the current passes, but at the same time caustic soda 
is found at the negative and sulphuric acid at the 
positive plate. The operation here is, Na 2 S0 4 is 
separated into Na 2 at the negative and S0 4 at the 
positive plate, but Na 2 decomposes water, — 

2Na+ 2H a O = 2NaHO + 2H, 

the 2H escaping from the negative plate. At the 
same time S0 4 at the positive plate decomposes water, 
liberating oxygen there : 

S0 4 +H 2 = H 2 S0 4 +0. 

Definitions of Terms. — The two plates forming 
the generator terminals are called electrodes. The 
positive plate by which the current enters the solution 
is called the anode. The negative plate by w r hich the 
current leaves the solution is called the cathode. The 
solution which is decomposed is called the electrolyte, 
and the operation is called electrolysis. The two 
parts into which the electrolyte is separated by the 
current are called ions. Thus Cu and S0 4 are the 
two ions when solution of CuS0 4 is the electrolyte. 



72 NOTES UPON ELECTRICAL MEASUREMENTS. 

Ions appear only at Electrodes. — The ions ap- 
pear only at their respective electrodes. No indica- 
tion of any separation is found elsewhere, if the elec- 
trolyte is continuous from electrode to electrode. 

Theory of electrolysis. 

Faraday's Laws- — Faraday demonstrated, first, 
that the quantity of an electrolyte decomposed in a 
given time is proportional to the current as measured 
by the tangent galvanometer; second, that the quan- 
tity of any ion set free per second by a given current 
is proportional to its chemical combining number. 
Thus, one ampere sets free in one second 

o.ooi 1 1 8 grammes of silver, 
0.0003279 " " copper, 
0.0003367 " " zinc, 
0.00001038 " " hydrogen, 
0.1 1 746 cc. of hydrogen. 

The combining numbers are: 



Silver l °7-7 

Copper 31.59 

Zinc 3 2 -44 

Hydrogen . . . 1 . 






Because of this exact relation, wh ? ch has been 
determined and verified by most careful experiments, 
the deposition of silver is taken as a means of stand- 
ardizing current-measuring instruments. 



CHEMICAL EFFECTS, 73 

Applications. — Electrotype. 

Electroplating: Gold, silver, nickel. 

Purification of metals by electro-deposition. 

Energy required for Electrolysis. — The decom- 
position of a chemical compound is in general work 
that can only be accomplished by the expenditure of 
energy. Now work done by an electric current has 
been shown to be IV = EC y where E is the E.M.F. 
or potential difference required to force the current 
C through the conductor or apparatus where the work 
is performed. For many chemical compounds the 
energy required for decomposition is well known, and 
since the current required is also known, the E.M.F. 
can be determined. For example, the combustion 
of one gramme* of hydrogen develops 34 calories 
= 141440 joules. To decompose sufficient water to 
produce one gramme of hydrogen will require the 
same energy. To produce one gramme of hydrogen 

per second would require - = 06,340 am- 

r ^ .00001038 * J ^ 

peres nearly. Hence the E.M.F. required is — ^— — 

= 1.47 volts, nearly. This is the count er-z\zoX.xo- 
motive force of the electrolyte, and the decomposition 
of water with the visible formation of gas requires 
that this E.M.F., and as much more as is necessary 
to force the current through the liquid against its 
resistance, be supplied. The counter-E.M.F. is in- 



74 NOTES UPON ELECTRICAL MEASUREMENTS. 

dependent of the strength of the current, while the 
E.M.F. required to overcome resistance is propor- 
tional to the current and equal to CR. For the 
economical decomposition of an electrolyte CR should 
be small in comparison to the counter-E.M.F. 
That is, R must be made as small as possible. 

When, as in most cases of electroplating, the anode 
is the same as the metal deposited, so that the com- 
position of the electrolyte remains unchanged, the 
energy developed by the union of the negative ion 
with the anode exactly counterbalances the energy 
required to decompose the electrolyte. Hence the 
energy required for the operation in such cases is only 
C 2 R, which goes to warm the electrolyte. 

PROBLEMS. 

(32) An apparatus for generating hydrogen and 
oxygen by means of the electric current, consisting of 
twelve decomposing cells arranged in series, furnishes 
12 cubic feet of oxygen per hour. What current is 
employed ? 

(33) Silver, copper, and zinc voltameters are 
arranged in series on one circuit. A current of 20 
amperes flows through them. How much metal per 
hour is deposited in each ? 

(34) To determine the constant of an ammeter, a 
current is passed through it and through a silver 
voltameter. A current giving a reading 25 deposits 






CHEMICAL EFFECTS. 75 

104.6 grms. of silver per hour. What is the galva- 
nometer constant ? 

(35) In an apparatus for electrical separation of 
silver, 30,000 (Troy) ounces of silver are deposited 
every 24 hours. What current is employed ? What 
horse-power, assuming the E.M.F. to be 2 volts ? 



76 NOTES UPON ELECTRICAL MEASUREMENTS. 



ELECTROMAGNETIC INDUCTION. 

It has been shown that the movement of a wire 
across the lines of force in a magnetic field produces 
an E.M.F. If this wire form part of a closed circuit, 
a current will be produced, and this consumes energy. 

Direction of Current. — The direction of the cur- 
rent produced is such that the movement of the con- 
ductor is opposed by the mutual action between the 
current and the field. 

Examples of electromagnetic induction. 

Currents result in all cases from a change in the 
number of lines of force threading through the circuit. 

Law of Lenz. — When a current is induced by any 
change whatever in the relations between a conductor 
and a magnetic field, that current is in such sense as 
to oppose the change that produces it. 

It follows that mechanical energy is consumed 
whenever a current is induced through the relative 
motion of a conductor and a magnetic field. From 
the law of conservation this energy must be the 
equivalent of the electrical energy developed. 

Electromotive Force Developed by Electromag- 
netic Induction. — Suppose a current C and an 



ELECTROMAGNETIC INDUCTION. J? 

E.M.F. E are induced by the movement of a wire in 
a magnetic field. The electrical energy developed in 
time t is ECt. To move the wire must, by the law 
of Lenz, require a force F, and if in time t the wire 
moves parallel to itself through a distance s, the work 
done is Fs. From the last paragraph 

Fs = ECt (37) 

It has been shown that the force F exerted by a 
current C upon a magnetic pole is (see equation (20)) 

CLp 



F = 



d' 



This is also the force which the pole exerts upon the 

P 
current, but -75, is the intensity of field at distance d 

from a pole whose strength is /. Calling this H, we 
have 

F=CLH. (38) 

This is the force exerted upon current C when placed 
in a magnetic field whose intensity is H. Substitut- 
ing this value of Fm (37), 

CLHs = ECt, 
or 

Z7 HLS f 

E = ~f~'> .... (39; 

But L is the length of the moving conductor and s 
the distance it moves parallel to itself. Ls is then the 



78 NOTES UPON ELECTRICAL MEASUREMENTS. 
area swept through by it, and HLs the total number 

TT T 

of lines of force cut by it. is then the number 

of lines of force cut per second. We have then the 
E.M.F. developed by a conductor moving in a mag- 
netic field is numerically equal to the rate of cutting 
of lines of magnetic force by that conductor. 

In order that the conductor of the above discussion 
may generate a current it must form part of a closed 
circuit. It is evident that the lines cut by the con- 
ductor are either added to or subtracted from the 
circuit of which the conductor forms a part." Hence, 
finally, the E.M.F. developed in a circuit by electro- 
magnetic induction is numerically equal to the rate at 
which lines of magnetic force are added to or sub- 
tracted from that circuit. If the lines of force here 
considered are c.g.s. lines, the E.M.F. is in c.g.s. 
units. To reduce to volts, divide by io 8 . 



ELECTROMAGNETISM. ?Q 



ELECTROMAGNETISM. 

It has been shown that a conductor carrying a cur- 
rent develops a magnetic field which may be repre- 
sented by lines forming closed curves around the 
conductor. 

In order that we may have a continuous current the 
conductor must be connected to the two terminals of 
a generator, and this, with the conductor, forms a 
closed electric circuit. 

The electric circuit and the lines of magnetic force 
due to it therefore form two interlinking closed 
curves. 

The Magnetic Circuit. — The intensity of the 
magnetic field set up by a current depends not only 
upon the strength of the current, but also upon the 
material that fills the space around the current and in 
which the magnetic lines must be formed. This 
material in which the magnetic lines are developed 
is considered as forming a magnetic circuit, akin to an 
electric circuit, and offering a greater or less opposi- 
tion, depending upon the nature and dimensions of 
the materials composing it, to the development of 



80 NOTES UPON ELECTRICAL MEASUREMENTS. 

magnetic lines, just as an electric circuit offers more 
or less resistance to the flow of the electric current. 

Magnetic Reluctance and Permeability. — The 
opposition offered by a magnetic circuit to the forma- 
tion in it of magnetic lines is called magnetic reluc- 
tance. Compare electrical resistance. The reluctance 
of any part of a magnetic circuit, like the resistance 
of any part of an electric circuit, is directly as its 
length and inversely as its cross-section. It is also 
proportional to the specific reluctance of the material. 
But this specific reluctance, unlike the specific resist- 
ance of materials forming electrical conductors, often 
varies greatly with the amount of flux. 

Magnetic permeability is the reciprocal of reluc- 
tance. It is akin to electrical conductivity. 

Carrying the analogy of the magnetic to the electric 
circuit still farther, the magnetism developed is often 
spoken of as a flow of magnetism around the circuit, 
or as a magnetic flux. The cause of magnetic flux is 
called magnetomotive force. 

The Electromagnet. — An electromagnet consists 
of a mass of iron wound with an electric conductor 
forming a coil or solenoid. When a current flows 
through this coil, a magnetomotive force is developed 

which can be shown to be equal to , where n is 

^ 10 

the number of turns in the coil and c is the current in 

amperes. The product ?ic is called the ampcrc-turus 



ELECTRO MA GNE TISM. 8 I 

of the coil. The magnetic flux produced by this 
magnetomotive force is 

~ ioP' 

where P is the reluctance of the magnetic circuit. 
The reluctance is derived from the dimensions and 
permeabilities of the different parts of the circuit. If 
a given part of the circuit has a length /, a cross- 
section A> and a permeability M> — that is, a specific 

i / 

reluctance -, — the reluctance of that part is -r—. 
M Am 

M for air is unity; for iron M varies from 3000 down 

to unity, depending upon the density of the magnetic 

flux through it. /* varies greatly also in different 

qualities of iron. It is, therefore, often necessary to 

determine its values for the particular quality of iron 

it is proposed to use in an electromagnet. 

Measurements of Permeability. — To find the per- 
meability of a specimen of iron, it is necessary to 
determine the magnetic flux corresponding to various 
magnetomotive forces. This is best done by deter- 
mining the E.M.F. induced in a small test-coil sur- 
rounding the test-piece, when this test-piece is 
subjected to the influence of a suddenly applied or 
suddenly reversed magnetomotive force. 

The test-piece may be made in the form of a ring 
which is uniformly wrapped with wire, through which 
an electric current can be sent to supply the magneto- 



82 NOTES UPON ELECTRICAL MEASUREMENTS. 

motive force. The small test-coil is formed by wrap- 
ping over a small portion of the ring a number of 
turns of fine wire. This is connected to the coil of a 
ballistic galvanometer. If now a current be sent 
through the magnetizing coil, magnetism is induced 
in the iron ring. This is equivalent to suddenly in- 
troducing a magnet into the test-coil, and induces in 
this a current which, acting for an instant upon the 
needle of the ballistic galvanometer, causes this to 
swing over a certain angle. The angle of this first 
swing is called the " throw " of the needle. It can 
be shown that, when the current is of short duration 
compared to the time of vibration of the needle, the 
quantity of electricity passing through the galvanom- 
eter coil is proportional to the sine of half the throw. 
The quantity of electricity flowing is 

where E is the E.M.F. developed in the test-coil, / 
the time it acts, and R the resistance of the test-coil 
circuit. But E is proportional to the rate of change 
of magnetic lines, and to the number of turns in the 
test-coil, or 

N n 
E = ltt- ho) 

where N x is the total change in magnetic flux, n l the 



ELECTROMAGNETlSM. 83 

number of turns in the test-coil, and t the time occu- 
pied in making the change. Hence 

^ io'R- 

It is usual in such measurements to reverse the 
current in the magnetizing coil. If N represent the 
total flux produced by the current in one direction, 
the total change due to reverse is N l = 2YV, and 

If K be the constant of the galvanometer, and the 
throw, we have 

Q = K sin \6 (42) 

Substituting in (41) and solving for N, 

._ io'^sin£0 , x 

N = ^ (43) 

If n be the number of turns in the magnetizing coil 
and c the current in amperes, the magnetomotive 
force is 

M =~^ • (44) 

If / be the length of the ring, that is, the mean cir- 
cumference, and A the area of its cross-section, the 
reluctance is 



84 NOTES UPON ELECTRICAL MEASUREMENTS. 
The total flux is then 



Hence 



M ^nncAfjt 
N= ~P = 10/ * 






N 
But -r is the magnetic flux per unit area, and is 

JT1. 

4.7Z71C 

generally represented by the capital letter B. j- 

is the magnetomotive force per unit length, and is 
represented by H. Hence 



and finally 



^ = JJ (46) 



lo'RKsm^d 10/ 

M = —s^r " x 4^- • • (47) 



The only quantity in the above formula whose 
determination presents any difficulty is K, the con- 
stant of the ballistic galvanometer. This is best 
determined by experiment. Connect in circuit with 
the galvanometer a coil of known number of turns, 
turn that coil through 180 in a known magnetic field, 
and note the deflection, 6. For example, let the coil 
lie flat upon a table where the vertical component of 
the earth's magnetic field is known. Pick up the coil 
and turn it quickly over. If V represent the vertical 



ELECTROMAGNETISM. 85 

intensity of the earth's field, and A the area of 
the coil, the number of lines passing through the 
coil is A V y and the total change of lines in turn- 
ing the coil over is 2 A V. In formula (43), for N 
substitute the value of A V\ for n A substitute the 
number of turns in the coil used; fori? substitute 
the resistance of the circuit including that coil and 
the galvanometer; for 6 substitute the observed de- 
flection ; and solve for K. 

The ring is not a convenient form for the test-piece 
for practical tests, as it is impossible to readily substi- 
tute one sample of iron for another. For practical 
tests a massive soft-iron forging in the form of a U is 
wrapped with the magnetizing coil. The test-piece, 
which is a straight round bar of much smaller cross- 
section, is clamped across the top of the U. Upon 
this test-piece the small test-coil is placed. 

Different specimens of iron may be readily com- 
pared by preparing straight round bars of the same 
dimensions and observing the effect of these upon a 
magnetic needle placed in a fixed position when the 
test-pieces are in turn subjected to the influence of a 
magnetizing coil. This is called the magnetometer 
method. It is not adapted to exact determinations. 



INDEX. 



PAGE 

Acceleration, Unit of 4 

Ammeter 26 

Ampere 23 

Ampere, International, Definition of 44 

Anode 71 

Arc Lighting 67 

Ballistic Galvanometer 57 

Batteries, Resistance of 51 

Bridge, Divided-metre 58 

Bridge, Wheatstone's 47 

Bridge-wire, Calibration of 60 

Cathode ... 71 

Chemical Effects of Current 70 

Clark's Cell 44, 62 

Conductivity 37 

Coulomb 27 

Current, Chemical Effects of 70 

Current, Heating Effects of 63 

Current Instruments, Tests of 60 

Current, Measurement of 53 

Currents, Mutual Action of 26 

Current, Unit of 23 

Difference of Potential, Unit of 32 

Differential Galvanometer 46 

Divided Circuits 38 

Divided-metre Bridge 58 

Dyne 6 

Earth's Magnetic Field, Horizontal Intensity of. ... 18 

Electric Current, Effects of 20 

Electric Quantity 27 

Electrolysis , , 71 

87 






88 INDEX. 

PAGE 

Electrolysis, Energy Required 73 

Electrolyte 71 

Electrolytes, Resistance of 49 

Electromagnet 80 

Electromagnetism 78 

Electromagnetic Induction 76 

Electromotive Force, Definition of 33 

Electromotive Force, Examples of 34 

Energy 7 

Erg 6 

Fall of Potential, Current Measured by 54 

Faraday's Laws 72 

Filaments, Incandescent, Proportioning of 66 

Force, Definition of 3 

Force, Measurement of 6 

Force, Unit of 6 

Fundamental Units 2 

Galvanometer, Ballistic 57 

Galvanometer, Constant of. 25 

Galvanometer, Differential, * 46 

Gravitation Units 8 

Heating Effects of the Current 63 

Heat Units 9 

Horizontal Intensity 15 

Incandescent Lighting 65 

Incandescent-lamp Filaments, Proportioning of 66 

Induction, Electromagnetic 76 

Inertia 3 

Insulation Resistance 37 

Insulation Resistance, Measurement of 4S 

Ions 71 

Ions Appear Only at Electrodes 72 

Joule, Definition of 8 

Kinetic Energy 7 

Lamp, Arc, Source of Light from 67 

Law of Lenz 76 

Length, Standard of 2 

Lines of Force 13 



i 



INDEX. 89 

I'AGE 

Magnetic Circuit , . . . 79 

Magnetic Flux 80 

Magneto-motive Force 80 

Magnetic Field , 12 

Magnetic Field Due to Current 20 

Magnetic Field, Graphical Representation 15 

Magnetic Field, Measurement of 16 

Magnetic Field, The Earth's 15 

Magnetic Field, Intensity of 14 

Mance's Method 52 

Mass, Standard of 2 

Matter, Distinctive Characteristics of 3 

Measurement of Current 53 

Measurement of Potential 55 

Measurement of Resistance 45 

Megohm. . 37 

Microhm 37 

Motion , 3 

Motion, Uniformly Varied 4 

Multiple-arc Arrangement 39 

Ohm 37 

Ohm, International 44 

Ohm's Law 37 

Permeability .... 79 

Permeability, Measurement of 81 

Potential 29 

Potential Energy 7 

Potential Instruments, Tests of 61 

Potential, Measurement of 55 

Potentiometer 56 

Practical Units 8 

Quantity, Unit of 27 

Reluctance 79 

Resistance 36 

Resistance, Measurement of 45 

Resistance Sets 45 

Resistance Sets, Tests of 58 

Resistance, Units of 37 

Standard of Length. . . , , ,,..,.. 2 



9° INDEX. 

PAGE 

Standard of Mass 2 

Specific Resistance 36 

Tangent Galvanometer 25 

Tests of Current Instruments 60 

Tests of Instruments 58 

Tests of Potential Instruments ... 61 

Tests of Resistance Sets 5S 

Unit Acceleration 4 

Unit Current 23 

Unit Difference of Potential 32 

Unit Electrical Quantity 27 

Unit Force 6 

Unit Magnetic Field 14 

Unit Magnetic Pole 13 

Unit Resistance 37 

Unit Velocity. 3 

Unit Work , 6 

Units, Arbitrary 2 

Units, C. G. S. System of I 

Units, Fundamental 2 

Units, Gravitation 8 

Units, Heat 9 

Units, International 44 

Units, Practical 8 

Units, Relations of 9 

Velocity, Unit of 3 

Volt 33 

Volt, International 44 

Voltmeter 57 

Watt, Definition of 8 

Wheatstone's Bridge 47 

Work, Measurement of 6 

Work, Unit of 6 



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Merriman and Brook's Handbook for Surveyors. . . .12ino, mor., 

Merriman's Geodetic Surveying 8vo, 

" Retaining Walls and Masonry Dams 8vo, 

Mosely's Mechanical Engineering. (Mahan.) 8vo, 

Nagle's Manual for Railroad Engineers 12mo, morocco, 

Pattou's Civil Engineering ,8vo, 

" Foundations 8vo, 

Rockwell's Roads and Pavements in France 12mo, 

Runner's Non-tidal Rivers ; 8vo, 

Searles's Field Engineering 12mo, morocco flaps, 

" Railroad Spiral 12mo, morocco flaps, 

Siebert and Biggin's Modern Stone Cutting and Masonry. . .8vo, 

Smith's Cable Tramways 4to, 

" Wire Manufacture and Uses 4to, 

Spalding's Roads and Pavements 12mo, 

" Hydraulic Cement 12mo, 

Thurston's Materials of Construction 8vo, 

* Trautwine's Civil Engineer's Pocket-book. ..12mo, mor. flaps, 

* " Cross-section Sheet, 

* *' Excavations and Embankments Svo, 

8 



25 00 


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2 50 


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* Train wine's Laying Out Curves 12mo, morocco, $2 50 

Waddell's De Pontibus (A Pocket-book for Bridge Engineers;. 

12mo, morocco, 

Wait's Engineering and Architectural Jurisprudence 8vo, 

Sheep, 

" Law of Field Operation in Engineering, etc 8vo. 

Warren's Stereotomy — Stone Cutting 8vo, 

Webb s Engineering Instruments 12mo, morocco, 

Wegmann's Construction of Masonry Dams 4to, 

Wellington's Location of Railways. . . 8vo, 

Wheeler's Civil Engineering 8vo, 

WolfTs Windmill as a Prime Mover . .8vo, 

HYDRAULICS. 

Water-wheels— Windmills— Service Pipe— Drainage, Etc. 
(See also Engineering, p. 6.) 
Bazin's Experiments upon the Contraction of the Liquid Vein 

(Trautwine) 8vo, 2 00 

Bovey's Treatise on Hydraulics 8vo, 4 00 

Coffin's Graphical Solution of Hydraulic Problems . . 12mo, 2 50 

Ferrel's Treatise on the Winds, Cyclones, and Tornadoes. . .8vo, 4 00 

Fuerte's Water and Public Health 12mo, 1 50 

Ganguillet&Kutter'sFlow of Water. (Heiing& Trautwine.). 8vo, 4 00 

Hazen's Filtration of Public Water Supply 8vo, 2 00 

Herschel's 115 Experiments 8vo, 2 00 

Kiersted's Sewage Disposal 12mo, 1 25 

Kirkwood's Lead Pipe for Service Pipe 8vo, 1 50 

Mason's Water Supply 8vo, 5 00 

Merriman's Treatise on Hydraulics. . 8vo, 4 00 

Nichols's Water Supply (Chemical and Sanitary) 8vo, 2 50 

Ruflner's Improvement for Non-tidal Rivers Svo, 1 25 

Wegmann's Water Supply of the City of New York 4to, 10 00 

Weisbach's Hydraulics. (Du Bois.) 8vo, 5 00 

Wilson's Irrigation Engineering 8vo, 4 00 

" Hydraulic and Placer Mining 12mo, 2 00 

Wolff's Windmill as a Prime Mover 8vo, 3 00 

Wood's Theory of Turbines Svo, 2 50 

MANUFACTURES. 

Aniline — Boilers— Explosives— Iron— Sugar — Watches— 
Woollens, Etc. 

Allen's Tables for Iron Analysis 8vo, 3 00 

Beaumont's Woollen and Worsted Manufacture 12mo, 1 50 

Bolland's Encyclopaedia of Founding Terms 12mo, 3 00 

9 



BollancTs The Iron Founder 12mo, 

" " " " Supplement 12mo, 

Booth's Clock and Watch Maker's Manual 12mo, 

Bouvier's Handbook on Oil Painting 12mo, 

Eissler's Explosives, Nitroglycerine and Dynamite 8vo, 

Ford's Boiler Making for Boiler Makers 18mo, 

Metcalfe's Cost of Manufactures 8vo, 

Metcalf 's Steel— A Manual for Steel Users 12mo, 

Reimann's Aniline Colors. (Crookes.) 8vo, 

* Reisig's Guide to Piece Dyeiug 8vo, 

Spencer's Sugar Manufacturer's Handbook 12mo, mor. flap, 

" Handbook for Chemists of Beet Houses. 

12mo, mor. flap, 

Svedelius's Handbook for Charcoal Burners 12mo, 

The Lathe and Its Uses 8vo, 

Thurston's Manual of Steam Boilers 8vo, 

Walke's Lectures on Explosives 8vo. 

West's American Foundry Practice 12mo, 

Moulders Text-book 12mo, 

Wiechmaun's Sugar Analysis 8vo, 

Woodbury's Fire Protection of Mills 8vo, 

MATERIALS OF ENGINEERING. 

Strength — Elasticity — Resistance, Etc. 
(See also Engineering, p. 6.) 

Baker's Masonry Construction 8vo, 

Beardslee and Kent's Strength of Wrought Iron 8vo, 

Bovey's Strength of Materials 8vo, 

Burr's Elasticity and Resistance of Materials 8vo, 

Byrne's Highway Construction 8vo, 

Carpenter's Testing Machines and Methods of Testing Materials. 

Church's Mechanics of Engineering — Solids and Fluids 8vo, 

Du Bois's Stresses in Framed Structures 4to, 

Hatfield's Trausverse Strains 8vo, 

Johnson's Materials of Construction 8vo, 

Lanza's Applied Mechanics. 8vo, 

Merrill's Stones for Building and Decoration 8vo, 

Merriman's Mechanics of Materials 8vo, 

11 Strength of Materials 12mo, 

Patton's Treatise on Foundations 8vo, 

Rockwell's Roads and Pavements in France 12mo, 

Spalding's Roads and Pavements 12mo, 

Thurston's Materials of Construction 8vo, 

10 



2 50 


2 50 


2 00 


2 00 


4 00 


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5 00 


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6 00 


5 00 


4 00 


2 50 


2 50 


2 50 


2 50 



5 00 


1 50 


7 50 


5 00 


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6 00 


.0 00 


5 00 


6 00 


7 50 


5 00 


4 00 


1 00 


5 00 


1 25 


2 00 


5 00 



Thurston's Materials of Engineering 3 vols., 8vo, $8 < ! 

Vol. I, Non-metallic 8vo, 2 00 

Vol. II., Iron and Steel 8vo : 3 50 

Vol. III., Alloys, Brasses, and Bronzes 8vo, 2 50 

Weyrauch's Strength of Iron and Steel. (Du Bois.) 8vo, 1 50 

Wood's Resistance of Materials 8vo, 2 00 

MATHEMATICS. 

Calculus— Geometry— Trigonometry, Etc. 

Baker's Elliptic Functions 8vo, 1 50 

Ballard's Pyramid Problem 8vo, 1 50 

Barnard's Pyramid Problem 8vo, 1 50 

Bass's Differential Calculus 12mo, 4 00 

Brigg's Plane Analytical Geometry 12mo, 1 00 

Chapman's Theory of Equations 12mo, 1 50 

Chessin's Elements of the Theory of Functious. 

Compton's Logarithmic Computations 12mo, 1 50 

Craig's Linear Differential Equations 8vo, 5 00 

Davis's Introduction to the Logic of Algebra 8vo, 1 50 

Halsted's Elements of Geometry ...8vo, 1 75 

11 Synthetic Geometry 8vo, 150 

Johnson's Curve Tracing 12mo, 1 00 

Differential Equations— Ordinary and Partial 8vo, 3 50 

" Integral Calculus 12mo, 1 50 

n " " Unabridged. 

" Least Squares 12mo, 1 50 

Ludlow's Logarithmic and Other Tables. (Bass.) 8vo, 2 00 

Trigonometry with Tables. (Bass.) 8vo, 3 00 

Mahan's Descriptive Geometry (Stone Cutting) .8vo, 1 50 

Merriman and TVood ward's Higher Mathematics 8vo, 5 00 

Merriman's Method of Least Squares 8vo, 2 00 

Parker's Quadrature of the Circle 8vo, 2 50 

Rice and Johnson's Differential and Integral Calculus. 

2 vols, in 1, 12mo, 2 50 

Differential Calculus 8vo, 3 00 

" Abridgment of Differential Calculus 8vo, 1 50 

Searles's Elements of Geometry 8vo, 1 50 

Totten's Metrology 8vo, 2 50 

Warren's Descriptive Geometry 2 vols., 8vo, 3 50 

' ■ Drafting Instruments 12mo, 1 25 

" Free-hand Drawing 12mo, 1 00 

11 Higher Linear Perspective 8vo, 3 50 

" Linear Perspective 12mo, 100 

fa Primary Geometry 12mo, 75 

11 



Warren's Plane Problems. 12mo, $1 25 

" Problems and Theorems 8vo, 2 50 

" Projection Drawing 12mo, 1 50 

Wood's Co-ordinate Geometry 8vo, 2 00 

'• Trigonometry 12mo, 1 00 

Woolf's Descriptive Geometry Royal 8vo, 3 00 

MECHANICS-MACHINERY. 

Text-books and Practical Works. 
(See also Engineering, p. 6.) 

Baldwin's Steam Heating for Buildings 12mo, 2 50 

Benjamin's Wrinkles and Recipes 12mo, 2 00 

Carpenter's Testing Machines and Methods of Testing 

Materials 8vo. 

Chordal's Letters to Mechanics 12mo, 2 00 

Church's Mechanics of Engineering 8vo, 6 00 

" Notes and Examples in Mechanics 8vo, 2 00 

Crehore's Mechanics of the Girder 8vo, 5 00 

Cromwell's Belts and Pulleys 12mo, 1 50 

Toothed Gearing 12mo, 150 

Comptou's First Lessons in Metal Working 12mo, 1 50 

Dana's Elementary Mechanics 12mo, 1 50 

Dingey's Machinery Pattern Making 12mo, 2 00 

Dredge's Trans. Exhibits Building, World Exposition, 

4to, half morocco, 10 00 

Du Bois's Mechanics. Vol. I., Kinematics 8vo, 3 50 

Vol. II.. Statics 8vo, 4 00 

Vol III., Kinetics 8vo, 3 50 

Fitzgerald's Boston Machinist 18mo, 1 00 

Flather's Dynamometers 12mo, 2 00 

Rope Driving 12mo, 2 00 

Hall's Car Lubrication 12mo, 1 00 

Holly's Saw Filing 18mo, 75 

Johnson's Theoretical Mechanics. An Elementary Treatise. 
(In the press.) 

Jones Machine Design. Part L, Kinematics 8vo, 1 50 

11 Part II., Strength and Proportion of 

Machine Parts. 

Lanza's Applied Mechanics 8vo, 7 50 

MacCord's Kinematics 8vo, 5 00 

Merriman's Mechanics of Materials 8vo, 4 00 

Metcalfe's Cost of Manufactures 8vo, 5 00 

Michie's Analytical Mechanics 8vo, 4 00 

Mosely's Mechanical Engineering. (Mahan.) 8vo, 5 00 

12 



Richards's Compressed Air 12mo, $1 50 

Robinson's Principles of Median ism 8vo, 3 00 

Smith's Press-working of Metals 8vo, 3 00 

The Lathe and Its Uses 8vo, 6 00 

Thurston's Friction and Lost Work 8vo, 3 00 

The Animal as a Machine , 12mo, 1 00 

Warren's Machine Construction 2 vols. , 8vo, 7 50 

Weisbach's Hydraulics and Hydraulic Motors. (Du Bois.)..8vo, 5 00 
" Mechanics of Engineering. Vol. III., Part I., 

Sec. I. (Klein.) 8vo, 5 00 

Weisbach's Mechanics of Engineering Vol. III., Part I., 

Sec. II (Klein.) 8vo, 5 00 

Weisbach's Steam Engines. (Du Bois.) 8vo, 5 00 

Wood's Analytical Mechanics 8vo, 3 00 

" Elementary Mechanics 12mo, 125 

" " " Supplement and Key 125 



METALLURGY. 

Ikon— Gold— Silver — Alloys, Etc. 

Allen's Tables for Iron Analysis 8vo, 

Egleston's Gold and Mercury 8vo, 

" Metallurgy of Silver Svo, 

* Kerl's Metallurgy — Copper and Iron 8vo. 

* " " Steel. Fuel, etc 8vo, 

Kunhardt's Ore Dressing in Europe 8vo, 

Metcalfs Sieel — A Manual for Steel Users 12mo, 

O'Diiseoll's Treatment of Gold Ores 8vo, 

Thurston's Iron and Steel 8vo, 

" Alloys Svo, 

Wilson's Cyanide Processes 12mo, 

MINERALOGY AND MINING. 

Mike Accidents — Ventilation — Ore Dressing, Etc. 

Barringer's Minerals of Commercial Value. .. .oblong morocco, 2 50 

Beard's Ventilation of Mines 12mo, 2 50 

Boyd's Resources of South Western Virginia Svo, 3 00 

" Map of South Western Virginia Pocket-book form, 2 00 

Brush and PenLeld's Determinative Mineralogy Svo, 3 50 

Chester's Catalogue of Minerals , . Svo 1 25 

4< " ' " .... papei , 5 

" Dictionary of the Names of Minerals 8vo, 3 00 

Dana's American Localities of Minerals 8vo, 1 00 

13 



3 00 


7 50 


7 50 


15 00 


15 00 


1 50 


2 00 


2 00 


3 50 


2 50 


1 50 



Dana's Descriptive Mineralogy. (E. S.) • • • -8vo, half morocco, 

" Mineralogy and Petrography (J.D.). 12rno, 

" Minerals and How to Study Them. (E. S.) 12mo, 

" Text-book of Mineralogy. (E. S.) 8vo, 

^Drinker's Tunnelling, Explosives, Compounds, and Rock Drills. 

4to, half morocco, 

Egleston's Catalogue of Minerals and Synonyms 8vo, 

Eissler's Explosives — Nitroglycerine and Dynamite 8vo, 

Goodyear's Coal Mines of the Western Coast 12mo, 

Hussak's Rock forming Minerals (Smith. ) 8vo, 

Ihlseng's Manual of Mining 8vo, 

Kuuhardt's Ore Dressing in Europe 8vo. 

O'Driscoll's Treatment of Gold Ores 8vo, 

Rosenbusch's Microscopical Physiography of Minerals and 

Rocks (Iddings ) . 8vo 

Sawyer's Accidents in Mines 8vo, 

Stockbridge's Rocks and Soils 8vo 

Walke's Lectures on Explosives 8vo. 

Williams's Lithology 8vo, 

Wilson's Mine Ventilation l6uio, 

" Hydraulic aud Placer Mining 12mo. 

STEAM AND ELECTRICAL ENGINES, BOILERS, Etc 

Stationary— Marine— Locomotive— Gas Engines, Etc. 

(See also Engineering, p. 6.) 

Baldwin's Steam Heating for Buildings 12mo, 

Clerk's Gas Engine , 1 2mo ; 

Ford's Boiler Making for Boiler Makers 18mo. 

Hemen way's Indicator Practice 12mo. 

Hoadley's Warm-blast Furuacc 8vo, 

Kneass's Practice and Theory of the Injector 8vo, 

MacCord's Slide Valve 8vo, 

* Maw's Marine Engines Folio, half morocco, 

Meyer's Modern Locomotive Construction 4to, 

Peabody and Miller's Steam Boilers 8vo, 

Peabody's Tables of Saturated Steam 8vo, 

" Thermodynamics of the Steam Engine 8vo, 

" Valve Gears for the Steam Engine 8vo, 

Pray's Twenty Years with the Indicator Royal 8vo, 

Pupin and Osterberg's Thermodynamics 12mo, 

Reagan's Steam and Electrical Locomotives 12mo, 

Rontgeii's Thermodynamics., (Du Bois.) 8vo, 

Sinclair's Locomotive Running 12mo, 

Thurston's Boiler Explosion 12mo, 

14 



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1 50 


3 50 


25 00 


2 50 


4 00 


2 50 


2 00 


4 00 


1 50 


2 00 


5 00 


7 00 


2 50 


4 00 


3 00 


1 25 



2 60 


4 00 


1 00 


2 00 


1 50 


1 50 


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18 00 


10 00 


4 00 


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2 50 


1 25 


2 00 


5 00 


2 00 


1 50 



Thurston's Engine and Boiler Trials 8vo, $5 00 

14 Manual of the Steam Engine. Part L, Structure 

and Theory 8vo, 7 50 

" Manual of the Steam Engine. Part II., Design, 

Construction, and Operation 8vo, 7 50 

2 parts, 12 00 

" Philosophy of the Steam Engine 12mo, 75 

" Reflection on the Motive Power of Heat. (Carnot.) 

12mo, 1 50 

" Stationary Steam Engines 12mo, 1 50 

" Steam-boiler Construction and Operation 8vo, 5 00 

Spangler's Valve Gears 8vo, 2 50 

Trowbridge's Stationary Steam Engines 4to, boards, 2 50 

Weisbach's Steam Engine. (Du Bois.) 8vo, 5 00 

Whitham's Constructive Steam Engineering 8vo, 10 00 

Steam-engine Design 8vo, 5 00 

Wilson's Steam Boilers. (Flather.) 12mo, 2 50 

Wood's Thermodynamics, Heat Motors, etc 8vo, 4 00 

TABLES, WEIGHTS, AND MEASURES. 

For Actuaries, Chemists, Engineers, Mechanics— Metric 
Tables, Etc. 

Adriance's Laboratory Calculations 12mo, 1 25 

Allen's Tables for Iron Analysis 8vo, 3 00 

Bixby's Graphical Computing Tables Sheet, 25 

Compton's Logarithms 12mo, 1 50 

Crandall's Railway and Earthwork Tables 8vo, 1 50 

Egleston's Weights and Measures 18mo, 75 

Fisher's Table of Cubic Yards Cardboard, 25 

Hudson's Excavation Tables. Vol, II 8vo, 1 00 

Johnson's Stadia and Earthwork Tables 8vo, 1 25 

Ludlow's Logarithmic and Other Tables. (Bass.) 12mo, 2 00 

Thurston's Conversion Tables 8vo, 1 00 

Totteu's Metrology 8vo, 2 50 

VENTILATION. 

Steam Heating — House Inspection — Mine Ventilation. 

Baldwin's Steam Heating 12mo. 2 50 

Beard's Ventilation of Mines 12mo, 2 50 

Carpenter's Heating and Ventilating of Buildings 8vo : 8 00 

Gerhard's Sanitary House Inspection Square 16mo, 1 00 

Mott's The Air We Breathe, and Ventilation 16nio, 1 00 

Reid's Ventilation of American Dwellings 12mo, 1 50 

Wilson's Mine Ventilation 8 16mo, 1 25 

15 



$5 00 


75 


1 50 


1 50 


2 00 


1 50 


4 00 


2 50 


1 00 


1 50 


3 00 



MISCELLANEOUS PUBLICATIONS. 

Alcott's Gems, Sentiment, Language Gilt edges, 

Bailey's The New Tale of a Tub 8vo, 

Ballard's Solution of the Pyramid Problem 8vo, 

Barnard's The Metrological System of the Great Pyramid. .8vo, 

Davis's Elements of Law 8vo, 

Emmon's Geological Guide-book of the Rocky Mountains. .8vo, 

FerreV s Treatise on the Winds 8vo, 

Haines's Addresses Delivered before r the Am. Ry. Assn. ..12mo. 
Mott's The Fallacy of the Present Theory of Sound. .Sq. 16mo, 

Perkins's Cornell University Oblong 4to, 

Ricketts's History of Rensselaer Polytechnic Institute 8vo, 

Rotherham's The New Testament Critically Emphasized. 

12mo, 1 50 
The Emphasized New Test. A new translation. 

Large 8vo, 2 00 

Totteu's An Important Question in Metrology 8vo, 2 50 

Whitehouse's Lake Moms Paper, '-25 

* Wiley's Yosemite, Alaska, and Yellowstone 4to, 3 00 

HEBREW AND CHALDEE TEXT=B00K5. 

For Schools and Theological Seminaries. 

Gesenius's Hebrew and Chaldee Lexicon to Old Testament. 

(Tregelles.) Small 4 to, half morocco, 5 00 

Green's Elementary Hebrew Grammar 12mo, 1 25 

Grammar of the Hebrew Language (New Edition ).8vo, 3 00 

Hebrew Cbrestomathy 8vo, 2 00 

Letteris's Hebrew Bible (Massoretic Notes in English). 

8vo ; arabesque, 2 25 
Luzzato's Grammar of the Biblical Chaldaic Language and the 

Talmud Babli Idioms 12mo, 1 50 

MEDICAL. 

Bull's Maternal Management in Health and Disease 12mo, 

Hammarsten's Physiological Chemistry. (Man del.) 8vo, 

Mott's ComureiiTjiQI)i^ratibility, and Nutritive Value of Food. 
£jg jW *^* ~ Large mounted chart, 

lludoflnan's Incompatibilities in Prescriptions 8vo, 

Steel's Treatise on the Diseases of the Ox 8vo, 

Treatise on the Diseases of the Dog 8vo, 

Woodhull's Military Hygiene 12mo, 

Worcester's Small Hospitals — Establishment and Maintenance, 
including Atkinson's Suggestions for Hospital Archi- 
tecture 12mo, 1 25 

16 



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